Nonlocal Exterior Calculus on Riemannian Manifolds

Home , Exterior calculus, Exterior derivative, Harley Flanders, Laplace operator

The Pennsylvania State University

The Graduate School

Department of Mathematics

NONLOCAL EXTERIOR CALCULUS

ON RIEMANNIAN MANIFOLDS

A Dissertation in

Mathematics

Thinh Duc Le

c 2013 Thinh Duc Le

Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Doctor of Philosophy

August 2013 ii

The dissertation of Thinh Duc Le was reviewed and approved* by the following:

Qiang Du

Verne M. Willaman Professor of Mathematics

Dissertation Adviser

Chair of Committee

Long-Qing Chen

Distinguished Professor of Material Sciences and Engineering

Ping Xu

Distinguished Professor of Mathematics

Mathieu Stienon

Associate Professor of Mathematics

Svetlana Katok

Director of Graduate Studies, Department of Mathematics

*Signatures are on ﬁle in the Graduate School. iii Abstract

Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries. Variations of these concepts have also been made over the years such as the discrete exterior calculus and the finite element exterior calculus. In this work, motivated by the recent studies of nonlocal vector calculus we develop a nonlocal exterior calculus framework on Riemannian manifolds which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal “interactions” (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Just as in the standard context, we introduce all necessary elements of exterior calculus such as forms, vector fields, exterior derivatives, etc. We point out the relationships between these elements with the known ones in (local) exterior calculus, discrete exterior calculus, etc. We also introduce nonlocal Hodge theory and its connections with existing works. iv Table of Contents

Acknowledgments ...... vi

Chapter 1. Introduction ...... 1 1.1 Local Exterior Calculus and Finite Element Exterior Calculus2 1.2 Discrete Exterior Calculus ...... 3 1.3 Existing Works in Nonlocal Vector Calculus ...... 4 1.4 Hodge Theory on Metric Spaces ...... 5 1.5 Details of this Dissertation ...... 7

Chapter 2. Nonlocal forms ...... 10 2.1 Oriented Simplices and Tuples ...... 10 2.2 Nonlocal Forms ...... 11 2.3 Nonlocal Exterior Derivative (D)...... 12 2.4 Codiﬀerential Operator (D∗)...... 17 2.5 NL Laplace-Beltrami Operator (∆) ...... 21 2.6 NL Hodge Operator (∗)...... 21 2.7 Nonlocal Wedge Product ∧ )...... 24 nl

Chapter 3. Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus ...... 25 3.1 Nonlocal Trivializations (λ)...... 25 3.2 Vector Fields ...... 27 3.3 NL Sharp (]) and Flat ([) Operators ...... 27 3.4 NL Vector Calculus ...... 31 v

3.5 Some special cases of NL vector calculus operators ...... 37

Chapter 4. Relationships to Local Geometry, Classical Vector Calculus and Discrete Vector Calculus ...... 43 4.1 Relationship between NL geometry and local geometry . . . . 43 4.2 Relationship between NL Vector Calculus and the Local One . 45 4.3 Relationship between NL Laplacian and discrete Laplacian . . 46

Chapter 5. NL Hodge Theory ...... 50 5.1 Hodge Theory for the Discrete Derivative ...... 50 5.2 Hodge Theory for the Nonlocal Exterior Derivative ...... 63

n Chapter 6. Another Model for Nonlocal Exterior Calculus in R ..... 67 6.1 Nonlocal Forms ...... 67 6.2 Nonlocal Exterior Derivative (D)...... 68 6.3 Codiﬀerential Operator (D∗)...... 69 6.4 NL Laplace-Beltrami Operator (revisit) ...... 70 6.5 Hodge Operator (∗)...... 70 6.6 Sharp (]) and Flat ([) Operators ...... 73 6.7 Vector Calculus (revisit) ...... 75 6.8 NL Wedge Product (Λnl)...... 76

Chapter 7. Ongoing and Future Works ...... 78 7.1 Ongoing Works ...... 78 7.2 Future Works ...... 78

References ...... 80 vi Acknowledgments

I would like to thank Professor Ping Xu and Professor Aissa Wade for giving me the chance to study at Penn State. Without their great support I would not be able to come to Penn State for my Ph.D. I would like to thank my advisor, Professor Qiang Du, for believing and trusting in my capability by giving me the opportunity to be one of his students and investing his research projects and grants in me. He has been always available and willing to help during my studies. I really appreciate that he is very patient with me and gives me great supports on both research and ﬁnance. Without his brilliant guidance this dissertation could not be done. I would like to thank Professor Ping Xu and Professor Mathieu Stienon from the Department of Mathematics and Professor Long-Qing Chen from the Depart- ment of Material Sciences and Engineering for helpful discussions and serving on my committee. I also would like to thank Doctor Tadele Mengesha for his help via a lot of discussions with him in my ﬁnal year at Penn State. I would like to give my appreciations to all the members in my research group for their helps and supports. Finally I would like to thank my family for their great supports especially when I am very far away from my home country. 1

Chapter 1

Introduction

Exterior calculus and differential forms are basic mathematical concepts that have been around for centuries [20]. They have made profound influence to the development of mathematics and they now can be learned from standard text books [33]. Variations of these concepts have also been made over the years. Highly suc- cessful examples include the discrete exterior calculus [10, 17] and the finite element exterior calculus [1] which are extensions to discrete spaces including piecewise linear complexes and finite element functions. They have proved to be useful in the development and analysis of finite element methods. More recently, motivated by the study of fractional diffusion processes and nonlocal electromagnetic media, the fractional exterior calculus has also been developed [9, 21, 34]. In this work, motivated by the recent studies of nonlocal calculus [16, 11], we develop a nonlocal (NL) exterior calculus framework which mimics many properties of the standard (local/smooth) exterior calculus. However the key difference is that nonlocal interactions (functions, operators, fields,...) are not required to be smooth. Also any point/particle can interact directly with any other point/particle in the studied domain (at least in principle). Since our work is partly related to local exterior calculus, discrete exterior calculus, existing works on nonlocal vector calculus and a L2 Hodge theory proposed by S. Smale et al. First of all we would like to give an literature overview on these theories which are relevant to our work. 2

1.1 Local Exterior Calculus and Finite Element Exterior Calculus

The exterior calculus of differential forms, also called Cartan’s calculus, is one whose geometric underpinning is the exterior algebra. It was born through a paper in 1899 by Elie´ Cartan ([6]). The first key ingredient of exterior calculus is differential forms. They are an approach to multivariable calculus that is independent of coordinates. Forms are local in the sense that they are defined pointwise: at any point x in a given manifold M, a p-form ω defines a skew-symmetric p-linear map p ω :(T M) → x x R and as an operator of x, ω is required to be smooth (here T M is the tangent space x to M at x). Forms can interact with each other via wedge product (exterior algebra). Differential forms provide a unified approach to defining integrals over curves, surfaces, volumes and higher dimensional (Riemannian) manifolds. Another key ingredient is vector fields. These are the dual of 1-forms via musical isomorphisms (sharp and flat operators). Besides these two operators, other key operators include exterior derivative, codifferential, Hodge star, Laplace- Beltrami, etc. We call “intrinsic properties” the relationships between key ingredients and key operators which are coordinate-free. Following the same procedure, we define our nonlocal exterior calculus with similar key ingredients and key operators. Our goal is to preserve as many intrinsic properties as possible. However due to nonlocality (almost all operators are integral operators), some intrinsic properties may not be achievable and some operators play less important roles (see Section 1.5 below). 3

Exterior calculus has many applications, especially in geometry, topology, partial differential equations (PDEs) and physics ([2],[13], [32]). Many partial differential equations (PDEs) are related to differential complexes, that is they can be rewritten using exterior calculus in neat forms. For example Maxwell’s equations can be written very compactly in geometrized units using forms, exterior derivative and Hodge star. From this point of view, the authors of [1, 2] develop the theory of Finite Element Exterior Calculus, which serves the study on numerical analysis and scien- tific computation of many PDEs. These PDEs are related to differential complexes but they are a fundamental component of problems arising in many mathematical models. This theory is developed to capture the key structures of the L2 de Rham complex and Hodge theory at the discrete level and to relate the discrete and continuous structures, in order to obtain stable finite element discretizations. The authors also develop an abstract Hilbert space framework (Hilbert complex), which captures key elements of Hodge theory and can be used to explore the stability of finite element methods. In our work we introduce the L2 nonlocal de Rham complex, which is a Hilbert complex. We use some results on Hilbert complexes in [2] to study this nonlocal de Rham complex.

1.2 Discrete Exterior Calculus

The authors of [10, 17] develop a theory of discrete exterior calculus (DEC) motivated by potential applications in computational methods for ﬁeld theories such as elasticity, ﬂuids, and electromagnetism. This discrete theory parallels the continuous (local) one in the sense that similar key ingredients and key operators are constructed (all in discrete forms) while some intrinsic properties are preserved. 4

The authors derive explicit formulas for some discrete differential operators in specific cases which are identical to the existing formulas in the literature. These formulas are proved to converge (in some sense) to their local (smooth) counterparts. In our work we first build the NL exterior calculus by following similar procedures as in the local or discrete exterior calculus. We also point out some relationships between the nonlocal exterior calculus and the local and the discrete ones. These relationships are either known in existing works (for example the convergence of NL operators to their local counterparts proved in [11], the use of the NL Laplacian as a bridge between the discrete one and the local one in [5]) or new (we propose new versions of operators in NL vector calculus in Chapter 3, a discretization of the NL Laplacian in Chapter 4). So our nonlocal exterior calculus is somewhere between the local one and the discrete one.

1.3 Existing Works in Nonlocal Vector Calculus

In [14] the authors defines some nonlocal operators including NL partial derivatives, NL gradient, NL divergence, etc. These basic operators are then used to define new types of flows and functionals for image and signal processing (and elsewhere). This framework can be viewed as an extension of spectral graph theory([7], [23]) and the diffusion geometry framework ([8], [24]) to functional analysis and PDE-like evolutions. However, the discussion in these papers is limited to scalar problems. In [11] the authors develop a framework for nonlocal vector calculus, including the definition of nonlocal divergence, gradient, and curl operators. Nonlocal analogs of several theorems and identities of the vector calculus for differential operators are 5 also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense. The operators of the nonlocal calculus are used to define volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators. Another application discussed in this paper is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations. Notice that in this work the authors define two different types of operators: point operators and two-point operators, where the latter ones are defined as adjoints of the former ones. In our work NL vector calculus is induced naturally from the nonlocal exterior calculus we develop just as in the standard context for the local one. That is vector calculus operators are defined using intrinsic relationships among key ingredients and key operators in exterior calculus (see Section 1.5 below). Thus we do not distinguish point operators and two-point operators (just as in local exterior calculus, there are only one gradient, one divergence and one Laplacian). Besides obtaining some similar basic operators and relationships among them as in [11], we also propose some new forms of these operators in Chapter 3 (Section 3.5). In some of these versions, two different horizon parameters are included, one is on the exterior derivative and the other is on the weight functions (also considered as on L2 spaces of forms). We see this as having a potential of reducing the singularity of kernels of NL operators.

1.4 Hodge Theory on Metric Spaces

The authors of [3, 30] develop a Hodge theory on metric spaces with two basic operators: a coboundary operator which is similar to the coboundary operator 6 of Alexander-Spanier cohomology ([31]) and a boundary operator, which is the L2- adjoint of the co-boundary operator. With these two operators the Hodge operator is defined, which is basically the Laplace-Beltrami operator. The authors develop an L2-Hodge theory and a Hodge theory at scale α. They also study α-harmonic forms, the α-cohomology and its relationship to the local de Rham cohomology. They also show that after rescaling, the α-Laplacian they define converges to the smooth one under some assumptions as α goes to 0. Notice that for the Hodge theory at scale α, the authors use the coboundary operator of Alexander-Spanier cohomology and the complexes depend on α. This scale α plays the same role as the horizon parameter in NL peridynamics and NL vector calculus in [27, 29, 12, 35]. Now the discrete exterior derivative mentioned in Section 1.1 is also the coboundary operator of Alexander-Spanier cohomology. In our work, this is a special case of the nonlocal exterior derivative. In general the nonlocal exterior derivative is an integral operator. The codifferential is also defined as the adjoint of the exterior derivative with respect to a weighted L2 inner product. The new approach in our work is that we embed the scale α (or the horizon parameter) into the weight functions and fix the complexes. Thus we only have one Hodge theory. Even though our work on Hodge theory is still in progress, it provides a different approach to the α-Hodge theory. Also note that the local (smooth) exterior derivative is an unbounded operator. Our work can handle this kind of situation and we can still obtain the Hodge decomposition (and Poincaré inequality). Due to our definition of the NL exterior derivative we also provide different versions of the NL Laplacian which converge to the smooth (local) one without rescaling. Also from the approximation theory point of view as in [2], one might want to 7 keep the L2- de Rham complex as a whole (not depend on any parameter) and then find a finite element approximation of this complex (there is an approximation parameter involving). The scaled cohomology in [3, 30] already depends on the scale α, so if one would like to approximate it while keeping α fixed, there could be more difficulty. Our goal is to apply the approximation theory for a general Hilbert complex already developed in [2] for our NL de Rham complex to get similar results as in local exterior calculus.

1.5 Details of this Dissertation

In chapter 2 we start with the discussion on nonlocal forms for oriented tuples defined on a Riemannian manifold. Nonlocal exterior derivative and codifferential operators are defined on forms, which are integral operators (in general). The L2 spaces here are weighted. From these two operators we define the Laplace- Beltrami, Hodge star and wedge product. Due to nonlocality the last two operators play a less important role. In chapter 3 we first discuss about the nonlocal trivialization. The idea of this comes from [26]. Then nonlocal operators showing the duality between 1-forms and vector fields are defined. These are sharp and flat. Based on these operators some standard operators in vector calculus are defined including gradient, divergence and Laplacian. We show some relationships between local and nonlocal operators. Especially we show that the local (smooth) Laplacian can be approximated by different forms of the nonlocal Laplacian, which is an integral operator. Besides a well known form of the NL Laplacian which involve a single integral we offer a new form which involves a triple integral. In chapter 4 we show the relationships between nonlocal geometry and local geometry, nonlocal vector calculus and the local and discrete ones, especially 8 about Laplacian operators. We show some examples where the NL trivialization can induce the usual Levi-Civita connection. Moreover there are examples of NL trivializations which do not correspond to any Levi-Civita connection. We also offer a discrete version of the NL Laplacian which can give convergence under some assumptions. In chapter 5 we introduce our work (in progress) on the nonlocal Hodge theory, which uses the Hilbert complex theory presented in [2]. This NL Hodge theory mimics and partly extends the results about the Hodge theory presented in [3] (the L2 Hodge theory results). This Hodge theory is believed to ”constitute a step towards understanding the geometry of vision” ([3]). In chapter 6 we introduce a different model for nonlocal exterior calculus n in the Euclidean space R , which has some advantages over the model presented in previous chapters (for example it preserves more intrinsic properties and uses a natural inner product). We redefine all key objects and operators using the natural L2 inner product. We cover some operators in vector calculus which already appear in previous chapters. Finally in chapter 7 we discuss about our ongoing and future works based on the work done in [2] for a general Hilbert complex. Recently NL calculus has been applied in studying image/signal processing and peridynamics. In [14] some NL operators (gradient, divergence, etc) are used to define some NL functionals. These functionals are called regularizing functionals, which replace the local notion of smoothness by the global notion of regularity. Based on these functionals some NL image/signal processing models are proposed. Due to the global regularity these models have been shown to have some advan- tage over similar local models such as the ability to detect and remove irregularities 9 from textures. In NL peridynamics theory ([27, 28, 29, 35]), there is no assumption on the differentiability of the displacement field, thus it has frequently been applied in the study of material failure. By studying NL operators systematically our framework can be used in these applications and for other purposes such as physical modeling of processes with NL behavior. The current notion of nonlocal exterior calculus also has many potential and natural applications. For instance, it can be used to study the approximations to classical (local) differential operators. 10

Chapter 2

Nonlocal forms

In this chapter we define nonlocal forms and basic operators acting on forms. These operators have same names and meaning as those in standard (local) exterior calculus such as differential operator, codifferential operator, Hodge operator. Our goal is to define these operators in a way such that they preserve as many intrinsic properties as possible. To begin our discussion, let M be a Riemannian manifold of dimension n with a metric tensor g. For each point x ∈ M we denote the tangent space at x by T and x the inner product at x by g . We also fix a volume form on M (with respect to x the metric tensor). From now on all integrals will be with respect to this volume form. In fact in this chapter we only need this volume form (so M here can be a metric space). On the next chapter we will make use of the (local) inner product g . x

2.1 Oriented Simplices and Tuples

Let e = (0, ..., 0) be the origin and e (for i = 1,...,n) be the ith unit vector 0 i n (consider as a point) in R . For a nonnegative integer p the standard p-simplex ∆p in n is the convex hull of {e , ..., e }. These points are vertices of the simplex. R 0 p Two ordering of the vertices are equivalent if they diﬀer from one another by an even permutation. If p > 0 then the orderings fall into two equivalence classes, each class is an orientation of the simplex. We simply write ∆p = [e , ..., e ] for 0 p the oriented simplex ∆p with the equivalence class of the ordering (e , ..., e ). 0 p 11

A singular p-simplex in the manifold M is a map s : ∆p → M (not necessary a one-to-one map). The points v = s(e ), i = 0,...,p are the vertices of the simplex in i i M. In order to deﬁne nonlocal exterior calculus we will actually only need the set of vertices (a tuple) of any simplex s in M and not the whole map s : ∆p → M. So we simply write s = [v , ..., v ] for an oriented p-tuple s in M. Basically a p-tuple 0 p is just an element of the set Mp+1.

Definition 2.1.1 We denote the real vector space generated by all oriented p- tuples in M by C (M). Each element of this space is called a p-chain and is p expressed as a finite formal sum of some oriented p-tuples (with coefficients in R).

Remark 2.1.2 If [v , ..., v ] is an oriented p-tuple then for any permutation σ in 0 p the symmetric group S we have p+1

[v , ..., v ] = sgn(σ)[v , ..., v ] (2.1) σ(0) σ(p) 0 p here sgn(σ) is the sign of the permutation σ.

2.2 Nonlocal Forms

p+1 Deﬁnition 2.2.1 A NL p-form is a map M → R, which is skew-symmetric. That is ω(v , ..., v ) = sgn(σ) ω(v , ..., v ) (2.2) σ(0) σ(p) 0 p for any permutation σ ∈ S . p+1 The vector space of all NL p-forms on M is denoted by Ωp (M). NL

Remark 2.2.2 1. A p-form can be extended linearly (over R) to a linear map C (M) → . So skew-symmetry property means that if the orientation of a p- p R tuple is changed then the value of a p-form on this p-tuple also switches to the 12 opposite sign. We refer to [17] (p. 29-30) for more discussion about forms.

2. A 0-form is just a function f : M → R. 3. In order to match up with the local exterior calculus, one can impose that all forms of order bigger than the dimension n is 0, that is Ωp (M) = {0} if p > n. NL

2.3 Nonlocal Exterior Derivative (D)

Consider an oriented p-tuple [v , ..., v ] with p > 0. This tuple has p+1 0 p faces which are oriented (p-1)-tuples where face i is [v , ..., vˆ , ..., v ]. Here the hat 0 i p means we omit that vertex.

Deﬁnition 2.3.1 The discrete exterior derivative d :Ωp (M) → Ωp+1(M) p NL NL is a linear operator deﬁned by

p+1 X i (d ω)(v , ..., v ) = (−1) ω(v , ..., vˆ , ..., v ) (2.3) p 0 p+1 0 i p+1 i=0 for any p-form ω and any (p+1)-tuple [v , ..., v ]. 0 p+1

Remark 2.3.2 The discrete exterior derivative is just the couboundary operator of Alexander-Spanier cohomology ([31]). We refer to [17] for discussion about the discrete derivative in discrete exterior calculus. Notice that d2 = 0, that is d ◦ d = 0 (we often omit the index p if there is no confusion). p+1 p

Now in order to deﬁne the NL exterior derivative, for each index p we introduce a map T satisfying the following properties (for all p = 0, 1,...): p i) T is a linear operator : Ωp (M) → Ωp (M), that is T preserves the skew- p NL NL p symmetry of forms. ii) T ◦ d = d ◦ T (T is a chain map). p+1 p p p 13

Let T = {T : p = 0, 1, ...} (T is a chain map by (ii) above). We see that the p set of all T satisfying (i) and (ii) forms a real vector space. We denote this space by T (M). Also note that the identity map I :Ωp (M) → Ωp (M) satisfies NL p NL NL (i) and (ii), thus I = {I : p = 0, 1, ...} belongs to T (M). We can also intro- p NL duce the multiplication on T (M) as composition of two maps with the same NL index. Then the set of all invertible elements of T (M) forms a group (note that NL if T ∈ T (M) is invertible then T −1 ∈ T (M)). We see that T (M) is a NL NL NL unitary associative algebra over . We usually write T instead of T if there is no R p confusion. We now define the NL differential operator D.

Deﬁnition 2.3.3 Given T ∈ T (M). The nonlocal exterior derivative D : NL p Ωp (M) → Ωp+1(M) is a linear operator deﬁned as NL NL

D = d ◦ T p p p

That is for a form ω ∈ Ωp (M), NL (Dω)(v , ..., v ) = Pp+1(−1)i (T ω)(v , ..., vˆ , ..., v ) 0 p+1 i=0 0 i p+1

Proposition 2.3.4D 2 = 0, that is D ◦ D = 0 p+1 p

Proof. By using property (ii) we see that

2 D ◦ D = d ◦ T ◦ d ◦ T = d ◦ d ◦ (T ) = 0. p+1 p p+1 p+1 p p p+1 p p

Since T :Ωp (M) → Ωp (M) is a linear operator we know that (from the p NL NL 14

Schwarz kernel theorem) its most general form is

Z 0 0 0 0 0 0 (T ω)(v , ..., v ) = K (v , ..., v ; v , ..., v ) ω(v , ..., v ) dv ...dv , p 0 p p 0 p Mp+1 0 p 0 p 0 p where K : M2p+2 → is some given kernel. Thus we have the following lemma: p R

Lemma 2.3.5 T belongs to T (M) iﬀ T preserves skew -symmetry of forms and NL the kernels K , p = 0, 1, ... satisfy the following identity p

p+1 X i 0 0 (−1) K (v , ..., vˆ , ..., v ; v , ..., v ) = p 0 i p+1 i=0 0 p

p+1 Z X i 0 0 0 0 (−1) K (v , ..., v ; v , ..., v , ..., v ) dv , (2.4) p+1 0 p+1 M i=0 0 p+1 p p+1 for all (v , ..., v ) and (v0 , ..., v0 ). Here on the i-th term of the sum on the right 0 p+1 0 p hand side, v0 is at position i starting from the semicolon (;). p+1 Proof. For any ω ∈ Ωp (M), we have NL (T d ω)(v , ..., v ) = p+1 p 0 p+1

p+1 Z 0 0 X i 0 0 0 0 = K (v , ..., v ; v , ..., v ) (−1) ω(v , ..., vˆ0, ..., v ) dv ...dv = p+1 0 p+1 0 p+1 i=0 0 i p+1 0 p+1

p+1 Z X i 0 0 0 0 0 0 0 = (−1) K (v , ..., v ; v , ..., v , ..., v ) ω(v , ..., v ) dv ...dv p+1 0 p+1 i=0 0 p+1 p 0 p 0 p+1 15

(v0 is at position i starting from the semicolon (;)). p+1 Also

p+1 X i Z 0 0 0 0 (d T ω)(v , ..., v ) = (−1) K (...vˆ ...; ...vˆ0...) ω(v , ..., vˆ0, ..., v ) dv ...dvˆ0...dv p p 0 p+1 p i i=0 i 0 i p+1 0 i p+1

p+1 Z X i 0 0 0 0 0 0 = (−1) K (v , ..., vˆ , ..., v ; v , ..., v ) ω(v , ..., v ) dv ...dv p 0 i p+1 i=0 0 p 0 p 0 p Comparing the results above we obtain the identity (2.4). We are still investigating the space T (M) to see if it has a canonical basis. NL In this manuscript, for computation and application purposes we usually use the following map T :Ωp (M) → Ωp (M) (and T = {T , p = 0, 1, ...}) p NL NL p

p Z Y 0 0 0 0 0 p (T ω)(v , ..., v ) = K(v , v ) ω(v , ..., v ) dv ...dv , ω ∈ Ω (M). 0 p p+1 i M i=0 i 0 p 0 p NL (2.5) 2 Here K : M → R is a kernel function which is non-negative, symmetric and satisﬁes Z K(x, y) dy = 1 (2.6) M One can see that T preserves both symmetry and skew-symmetry of functions on Mk+1. Also

p+1 0 0 Y 0 K (v , ..., v ; v , ..., v ) = K(v , v ) p+1 0 p+1 i 0 p+1 i=0 i and we see that

Z 0 0 0 0 0 K (v , ..., v ; v , ..., v ) dv = K (v , ..., vˆ , ..., v ; v , ..., vˆ0, ..., v ) p+1 0 p+1 p 0 i p+1 M 0 p+1 i 0 i p+1 16

(i = 0, 1,...). Thus T satisﬁes the identity (2.4) and it belongs to T (M). NL

Remark 2.3.6 We now list a few cases of interest of the operator D depending on the kernel K: 1. The kernel K is the Dirac delta function on M: K(x, y) = δ (y). In this case x the operator T is just the identity map and the NL diﬀerential operator is the discrete one (times a constant). n 2. M is R and the kernel K is a Gaussian kernel of the form

2 1 ||x−y|| − 4δ K(x, y) = n e , (4πδ) 2 here δ is a positive constant and ||x − y|| is the Euclidean distance between x n and y (notice that condition (2.6) holds and K is the heat kernel on R ). In this case the operator D is truly an integral operator. For 0-forms (functions) n on R the operator T is the same as the well known Poisson transform (Gauss transform/Gauss - Weierstrass transform is a special case when δ = 1). For a general Riemannian manifold M we can replace the kernel above by the heat kernel on M. The condition (2.6) means that the manifold M is required to be stochastically complete (see [15]). For example any compact Riemannian manifold is stochastically complete. n 3. M is R and the kernel K has the form   C(δ)  s if ||x − y|| ≤ δ K(x, y) = ||x−y||  0 otherwise Again δ is a positive constant and C(δ) is the normalized constant such that condition (2.6) holds. The exponent s can be chosen in interval [0, n − 1]. In this case D is also a truly integral operator. 17

2.4 Codiﬀerential Operator (D∗)

In local exterior calculus the codiﬀerential operator is the adjoint of the exterior derivative with respect to the standard inner product corresponding to the (local) volume form on the manifold M. Here ﬁrst of all we assume that for each index p, Ωp (M) is equipped with an inner product (, ) (we will specify one NL p later).

Definition 2.4.1 i) Given T ∈ T (M), the (formal) adjoint of T is a linear NL operator T ∗ :Ωp (M) → Ωp (M) defined by (T ω , ω ) = (ω ,T ∗ω ) , for any NL NL 1 2 p 1 2 p ω , ω ∈ Ωp (M)(p = 0, 1, ...). 1 2 NL ii) The (formal) adjoint of the discrete derivative d is a linear operator d∗ : p−1 Ωp (M) → Ωp−1(M) defined by (dω, η) = (ω, d∗η) , for any ω ∈ Ωp−1(M), η ∈ NL NL p p−1 NL Ωp (M)(p = 0, 1, ...). NL iii) The codifferential operator D∗ is the (formal) adjoint of the NL exterior derivative D. That is D∗ :Ωp (M) → Ωp−1(M) is a linear operator such p−1 NL NL that ∗ (Dω, η) = (ω, D η) (2.7) p p−1 for any ω ∈ Ωp−1(M), η ∈ Ωp (M). NL NL

Remark 2.4.2 1. The codifferential operator can be computed by D∗ = T ∗ ◦d∗ in some case (at least when d is bounded, see [36]). 2. We refer to [19] (p. 85-86) for the definition of the codifferential operator in local exterior calculus. 3. Because D2 = 0 we also have (D∗)2 = 0. 18

We now introduce a weighted L2 inner product on Ωp (M) for each index p. NL Weight functions: For each integer p = 0,...,n we ﬁx a function W : Mp+1 → p R such that: i) W (v , ..., v ) ≥ 0 for any (v , ..., v ) ∈ Mp+1 p 0 p 0 p ii) W is symmetric, that is W (v , ..., v ) = W (v , ..., v ) for any permu- p p σ(0) σ(p) p 0 p tation σ ∈ S . p+1 When p = 1 there are a few cases of interest for W . The trivial case is W ≡ 1. 1 1 The other cases are similar to examples 2 and 3 in Remark 2.3.6 when M is a n domain in R : W has a Gaussian form 1

2 ||x−y|| − W (x, y) = C(δ)e δ , 1 or   C(δ)  s if ||x − y|| ≤ δ W (x, y) = ||x−y|| 1  0 otherwise The constant C(δ) is chosen such that the following condition holds

Z 2 ||z|| W (z) dz = n (2.8) n 1 R Here z = x − y and W is considered as a function of z. The exponent s can be 1 chosen in interval [0, n + 1] We refer to [14] (p. 1010-1011) for more discussion about W in n. 1 R For a general manifold M one can use the heat kernel as in Remark 2.3.6 or the 2 d(x,y) − Gaussian form K(x, y) = C(δ)e δ , where d(x, y) is the geodesic distance on M and C(δ) satisﬁes a condition similar to (2.8). 19

When p > 1 we usually deﬁne W (v , ..., v ) = Q W (v , v ). p 0 p 0≤i

Deﬁnition 2.4.3 For ω, η ∈ Ωp (M) we deﬁne the (W -weighted) L2-product of NL p these two forms as

Z (ω, η) = ω(v , ..., v ) η(v , ..., v ) W (v , ..., v ) dv ...dv (2.9) 0 p 0 p p 0 p 0 p Mp+1

(again as mentioned at the beginning of this Chapter, all integrals are with respect to the ﬁxed volume form given by the metric g).

Remark 2.4.4 1. If W is positive everywhere (for example when W has a p p Gaussian form) then (, ) is obviously an inner product in the L2 sense. If this is not the case (for example when W is a cut-off function), in order to make (, ) an 1 inner product in the L2 sense one can have two options: i) Redefine forms as following: for any form ω ∈ Ωp (M), ω(v , ..., v ) = 0 if NL 0 p W (v , ..., v ) = 0 (this is well-defined because forms are skew-symmetric and p 0 p weight functions are symmetric). ii) Define an equivalence relation between any two p-forms ω, η as following: ω ∼ η iff ω(v , ..., v ) = η(v , ..., v ) whenever W (v , ..., v ) 6= 0. Then define 0 p 0 p p 0 p Ωp (M) as the space of all equivalence classes of p-forms. NL 2. In [3] the weight function W is embedded into the definition of the exterior 1 derivative thus the authors only use the standard L2 product (W ≡ 1 for all p) p

We also deﬁne L2(Ωp (M)) as the space of all p-forms ω such that ||ω|| < NL ∞, here ||.|| is the norm corresponding to the inner product above. This is a weighted L2 space and so it is a Hilbert space (L2 space of a measure on M) 20

Proposition 2.4.5 The codiﬀerential operator D∗ corresponding to the operator T in (2.5) has an explicit formula as following: for any p-form ω,

p ∗ (−1) (p + 1) (D ω)(v , ..., v ) = × 0 p−1 W (v , ..., v ) p−1 p−1 0 p−1

p−1 Z Y 0 0 0 0 0 0 0 K(v , v ) W (v , ..., v ) ω(v , ..., v ) dv ...dv (2.10) p+1 i p M i=0 i 0 p 0 p 0 p

(Of course D∗ ≡ 0 by default). −1 In particular (when K is the Dirac delta function):

p ∗ (−1) (p + 1) Z (d ω)(v , ..., v ) = W (v , ..., v ) ω(v , ..., v ) dv . 0 p−1 W (v , ..., v ) p 0 p 0 p p p−1 p−1 0 p−1 M

Proof. From the formula of D and the identity (2.7), by renaming variables inside integrals, one can obtain

∗ 1 (D ω)(v , ..., v ) = × 0 p−1 W (v , ..., v ) p−1 0 p−1 p Z X i 0 0 0 0 0 0 0 0 0 (−1) K(v , v )...K(v , v )...K(v , v ) W (v , ..., v ) ω(v , ..., v ) dv ...dv 0 i p−1 p i=0 0 i+1 p 0 p 0 p 0 p Here each product inside the sum always has p factors in order from v to v , 0 p−1 v0 to v0 and the variable v0 is omitted. 0 p i Since ω is skew-symmetric and W is symmetric, all (p+1) terms inside the integral p are equal and the term containing Qp−1 K(v , v0) is associated with (−1)p. Hence i=0 i i we obtain the formula (2.10) above.

Remark 2.4.6 One can also deﬁne the operator D∗ using the formula (2.10) then prove that it is formally adjoint to D. 21

2.5 NL Laplace-Beltrami Operator (∆)

Deﬁnition 2.5.1 As in standard context the NL Laplace-Beltrami operator ∆ : Ωp(M) → Ωp(M) is deﬁned as

∗ ∗ ∗ ∗ ∆ = DD + D D (= D D + D D ) p−1 p−1 p p

2.6 NL Hodge Operator (∗)

As usual the Hodge operator is a linear operator ∗ :Ωp (M) → Ωn−p(M) NL NL (we assume that Ωp (M) = {0} for p > n). NL Here we deﬁne this operator as following: for any p-form ω, (∗ω)(v , ..., v ) = 0 n−p

Z K (v , ..., v , v , ..., v ) ω(v , ..., v ) dv ...dv p 0 n−p n−p+1 n+1 n−p+1 n+1 n−p+1 n+1 Mp+1

The kernel K is a given function which is skew-symmetric with respect to (v , ..., v ) p 0 n−p and also skew-symmetric with respect to (v , ..., v ). n−p+1 n+1 We would like to ﬁnd condition(s) on the kernel functions such that the Hodge operator can maintain some intrinsic properties. One property we would like to have is ∗ p D ∗ ω = (−1) ∗ D ω for any p-form ω. 22

Lemma 2.6.1 Suppose that the diﬀerential operator D corresponds to T in (2.5). If the kernels K (p = 0, ..., n) are chosen such that the following condition p

n+1 Z Y 0 0 0 0 0 (p+2) K(v , v ) K (v , ..., v , v , ..., v ) dv ...dv p+2 i p+1 0 n−p−1 M i=n−p+1 i n−p n+1 n−p n+1

(−1)n (n−p+1) = × W (v ,...,v ) n−p−1 0 n−p−1

n−p−1 Z Y 0 0 0 0 0 0 0 K(v , v ) W (v , ..., v ) K (v , ..., v , v , ..., v ) dv ...dv n−p+1 i n−p p n−p+1 n+1 M i=0 i 0 n−p 0 n−p 0 n−p is satisﬁed for all p = 0, ..., n − 1 and all (v , ..., v , v , ..., v ) then 0 n−p−1 n−p+1 n+1 the identity ∗ p D ∗ ω = (−1) ∗ D ω holds for any p-form ω.

Proof. We will show direct calculation using the formulas of D and D∗ in Sections 2.3 & 2.4. On one hand (∗D ω)(v , ..., v ) = 0 n−p−1

n+1 X i−(n−p) Z (−1) K (v , ..., v , v , ..., v )(T ω)(v , ..., vˆ , ..., v ) p+1 0 n−p−1 n−p n+1 n−p i n+1 i=n−p

dv ...dv n−p n+1

Since K is skew-symmetric with respect to the second component (v , ..., v ), p+1 n−p n+1 by renaming variables we can see that all terms in the sum above are equal and thus 23

(∗D ω)(v , ..., v ) = (p + 2) × 0 n−p−1

n+1 Z Y 0 0 0 K(v , v ) K (v , ..., v , v , ..., v ) ω(v , ..., v ) i p+1 0 n−p−1 n−p n+1 i=n−p+1 i n−p+1 n+1

0 0 dv ...dv dv ...dv n−p n+1 n−p+1 n+1

On the other hand

n ∗ (−1) (n − p + 1) (D ∗ ω)(v , ..., v ) = × 0 n−p−1 W (v , ..., v ) n−p−1 0 n−p−1

n−p−1 Z Y 0 0 0 0 0 0 0 0 0 K(v , v ) W (v , ..., v ) K (v , ..., v , v , ..., v ) ω(v , ..., v ) i n−p p i=0 i 0 n−p 0 n−p n−p+1 n+1 n−p+1 n+1 0 0 dv ...dv 0 n+1

Rename (v0 , ..., v0 ) to (v , ..., v ). Comparing two expressions n−p+1 n+1 n−p+1 n+1 above we see that in order for the identity

∗ p D ∗ ω = (−1) ∗ D ω to hold for any p-form ω the condition in the statement of the lemma is suﬃcient.

Remark 2.6.2 In local exterior calculus, the Hodge operator (*) is an isomorphism at each point on M (between the p-th and (n-p)-th exterior products of the cotangent space at the given point, these vector spaces have ﬁnite dimensions). Moreover ∗∗ = (−1)p(n−p) , thus the inverse of ∗ :Λp(T ∗M) → Λn−p(T ∗M) is x x x (−1)p(n−p)∗ :Λn−p(T ∗M) → Λp(T ∗M). x x x 24

Here due to the nonlocality there is no guarantee that the NL Hodge still possesses these properties.

2.7 Nonlocal Wedge Product ∧ ) nl

In order to deﬁne the wedge product of NL forms we can use the deﬁnition proposed by Castrillon-Lopez for discrete exterior calculus (see [10], p. 49 or [17], p. 74-75).

Deﬁnition 2.7.1 The wedge product of a p-form ω and a q-form η is a (p+q)-form given by

1 X (ω∧ η)(v , ..., v ) = sgn(σ) ω(v , ..., v ) η(v , ..., v ) nl 0 p+q (p + q + 1)! σ(0) σ(p) σ(p) σ(p+q) σ∈S p+q+1

We can verify that (ω∧ η) is indeed a form (it is skew-symmetric) and satisﬁes nl the following properties: i) Anti-commutativity (ω∧ η) = (−1)pq (η∧ ω) nl nl ii) Leibniz rule d (ω∧ η) = (d ω)∧ η + (−1)p ω∧ (dη) nl nl nl iii) Associativity for closed forms For a p-form ω, a q-form η and a r-form γ such that dω = 0, dη = 0, dγ = 0, we have

(ω∧ η)∧ γ = ω∧ (η∧ γ) nl nl nl nl

We refer to [10] (p. 17-21 and p. 49) for the proof.

Remark 2.7.2 The wedge product deﬁned here only works for the discrete derivative d. 25

Chapter 3

Nonlocal Riemannian Geometry. Relationship between Forms and Vector Fields. Vector Calculus

In this chapter we introduce NL trivializations on the manifold M which could be reduced to the Levi-Civita connection in some special case (see next chapter). We also define NL vector fields besides the usual vector fields on M. We then define NL operators showing the relationship between (NL) forms and vector fields just as in the standard context. With the help of these operators we can construct (NL) vector calculus.

3.1 Nonlocal Trivializations (λ)

As mentioned in [26] a NL trivialization is a means to compare vector ﬁelds at any two points on manifold M directly, without any primary notion of inﬁnitesimal transport of vectors or the accompanying path-dependent parallel transport.

Deﬁnition 3.1.1 A nonlocal trivialization λ is a map which corresponds any or- dered pair of points (x, y) a linear isomorphism λ : T → T such that: xy x y i) λ is the inverse of λ : λ = λ−1 yx xy yx xy ii) λ : T → T is the identity map xx x x iii) λ preserves the inner products on T and T xy x y

Remark 3.1.2 1. We refer to [26] for more discussions about NL trivialization(s). The author in that manuscript only imposes condition (i) for a nonlocal trivialization. 26

2. There is no requirement about the continuity or smoothness of the NL trivialization. n 3. Consider the case when the manifold M is the Euclidean space R . The trivial NL trivialization on n is just the identity map λ = Id : n → n for any pair R xy R R (x, y). An example of a nontrivial NL trivialization can be obtained as follows: for any pair (x, y) with x 6= y, λ : n → n is a given reflection (not necessary in a xy R R hyperplane). Another example: for any pair (x, y) with x 6= y, λ : n → n is the reflection xy R R in the hyperplane through the origin, orthogonal to the vector (x − y). 2 3 4. Consider another example when M is the two-dimensional sphere S in R . 2 One can construct two NL trivializations on S as follows: First of all for each x ∈ 2, T is the plane through the origin O ∈ 3 and perpen- S x R dicular to x (so T = T ). If x = ± y then λ is the identity map Id : T → T . −x x xy x x If x 6= ± y there are two planes bisecting the angle ∠Oxy and its supplementary. Define Ref to be the reflection in one of these planes and let λ be the restriction xy of Ref to T . One can verify that (x − y) and (x + y) are the normal vectors to x the two planes above so

< v|x ± y > λ : T → T , λ (v) = v − 2 (x ± y), v ∈ T (3.1) xy x y xy ||x ± y||2 x

3 (< | > and ||.|| are the Euclidean inner product and norm in R ). It is clear that λ is orthogonal and λ = λ−1. xy yx xy From now on we assume that the manifold M is equipped with a NL trivialization λ. We call the geometry induced on M by a NL trivialization as NL 27

Riemannian geometry. Some of them will be discussed in subsequent sections. We refer to [26] for more details on NL diﬀerential geometry discussed there.

3.2 Vector Fields

We refer to a usual vector field u : M 3 x 7→ u(x) ∈ T as a point vector x field. We define a two-point (or NL) vector field V as a map

V : M × M 3 (x, y) 7→ V (x, y) ∈ T x

Deﬁnition 3.2.1 A two-point vector ﬁeld V is called i) λ-symmetric if λ (V (x, y)) = V (y, x) xy ii) λ-skew-symmetric if λ (V (x, y)) = −V (y, x) xy here λ is a given NL trivialization on M as in Section 3.1 .

Since the NL trivialization λ is given from now on we will only write symmetric/skew- symmetric instead of λ-symmetric/λ-skew-symmetric.

Remark 3.2.2 1. Again there is no requirement about continuity or smoothness of vector fields here. 2. Two-point vector fields can be created from point-based vector fields by using NL trivialization λ. For example if u : x 7→ u(x) ∈ T is a point-based vector x field then V :(x, y) 7→ V(x,y) = λ (u(y)) ∈ T is a two-point vector field. Also yx x V (x, y) = u(x)− λ (u(y)) is a two-point vector field which is skew-symmetric. yx

3.3 NL Sharp (]) and Flat ([) Operators

As in the standard context, sharp and flat are operators showing relationships between 1-forms and NL (two-point) vector fields. First of all we start with 28 a general definition for each of these operators and then we will use a specific one for computation and application purposes.

Definition 3.3.1 i) The sharp operator (]) is a linear operator which maps any 1-form ω to a NL vector field ω] defined by

] Z ] 0 0 00 00 0 0 00 00 0 0 00 00 ω (x, y) = σ (x, y, x , y , x , y ) ω(x , y ) λ (V (x , y )) dx dy dx dy . x00x 0] M4

Here σ] : M6 → is a given kernel and V is a given NL vector field (λ is the R 0] NL trivialization). ii) The flat operator ([) is a linear operator which maps any two-point vector field V to a 1-form V [ defined by

[ Z [ 0 0 00 00 0 0 00 00 0 0 00 00 V (x, y) = σ (x, y, x , y , x , y ) g V (x , y ), λ (V (x , y )) dx dy dx dy . x0 x00x0 0[ M4

Here σ[ : M6 → is a given kernel and V is a given NL vector ﬁeld . R 0[

Remark 3.3.2 1. Sharp and flat are linear operators between Ω1 (M) and the NL (real) vector space generated by all NL vector fields on M. 2. Note that ω](x, y) ∈ T thus in the definition (i) we need to use the NL x trivialization so that λ (V (x00, y00)) ∈ T (similarly for the definition (ii)). x00x 0 x

For the relationship between sharp and flat we would like to preserve the following properties just as in standard context and thus put some constraints on the given data (σ], σ[,V ,V ): 0] 0[ 1) V [ has to be a 1-form (skew-symmetric function on M2). 2) (ω])[ = ω for any 1-form ω and (V [)] = V for any NL vector field V . That is sharp and flat are the inverse of each other. However since 1-forms are scalar 29 functions and vector fields have dimension n, two identities above may not be achievable simultaneously. 3) g ω](x, y),V (x, y) = ω(x, y)V [(x, y) for all ω, V and (x, y). If this identity x happens we define this quantity as the pairing < ω, V > of ω and V at (x, y). Also < V [,V > (x, y) = g (V (x, y),V (x, y)) for any two vector fields V ,V . 1 2 x 1 2 1 2 Note that due to nonlocality we may not be able to preserve all properties as in local exterior calculus.

We now use a simple form of sharp and flat for computation and application purposes. First of all since 1-forms are skew-symmetric, their dual (NL vector fields) must have a similar property. Here we will define the dual of a 1-form as a symmetric NL vector field. To do that we fix a skew-symmetric vector field V :(x, y) 7→ V (x, y) ∈ T such that 0 0 x g V (x, y),V (x, y) = 1 if x 6= y (this means V (x, y) 6= 0 if x 6= y ). Here x 0 0 0 we assume that such a vector field exists on M (note that there is no requirement about continuity or smoothness). n For example if M is R with the trivial NL trivialization one can take V (x, y) = (y − x)/||y − x|| if x 6= y (and V (x, x) = 0). 0 0 Another example: consider the sphere 2. We write x ∈ 2 as x = (x , x , x ) S S 1 2 3 and let   (0, −x , x ), if x = 0  3 2 1 u (x) = 0 1 (−x , x , 0), if x 6= 0 r 2 1 1  x2+x2  1 2 Now we define 30   x × y, if x 6= ± y   V (x, y) = u (x), if x = −y 0  0   0, if x = y 3 (Here × is the cross product in R ). One can check that V (x, y) ∈ T and with the NL trivialization(s) defined in 0 x Section 3.1 for 2, λ (V (x, y)) = −V (y, x) and V (x, y) 6= 0 if x 6= y. S xy 0 0 0

Definition 3.3.3 i) The pairing between a 1-form ω and a two-point vector field V is defined as < ω, V > (x, y) = ω(x, y) g V (x, y),V (x, y) . x 0 ii) The sharp operator ] maps any 1-form ω to a symmetric two-point vector field ω] defined by ω](x, y) = ω(x, y)V (x, y). 0 iii) The flat operator [ maps any symmetric two-point vector field V to a 1-form V [ defined by V [(x, y) = g V (x, y),V (x, y) . x 0

Lemma 3.3.4 i) The pairing < ω, V > is bilinear and non-degenerate with respect to ω. ii) ω] is a symmetric vector ﬁeld.

iii) V [ is a 1-form. iv) < ω, V > (x, y) = g ω](x, y),V (x, y) = ω(x, y)V [(x, y)∀(x, y), for any x 1-form ω and any symmetric vector ﬁeld V . v) (ω])[ = ω, for any 1-form ω.

Proof. i) It is clear that if < ω, V >= 0 for any V then ω = 0. ii) λ ω](x, y) = ω(x, y) λ V (x, y) = − ω(y, x) − V (y, x) = xy xy 0 0 ω](y, x). iii) V [(y, x) = g V (y, x),V (y, x) = g λ (V (x, y)), −λ (V (x, y)) y 0 y xy xy 0 = −g V (x, y),V (x, y) = −V [(x, y) x 0 31

(because λ preserves the inner products on T and T ). xy x y iv) This is obvious. v) Note that g V (x, y),V (x, y) = 1 if x 6= y so (ω])[ = ω. x 0 0

Remark 3.3.5 1. The sharp and flat in Definition 3.3.3 are a special case of Definition 3.3.1 when we choose σ] and σ[ as a product of the Dirac delta function and V = V = V . 0] 0[ 0 2. The pairing < ω, V > is not non-degenerate with respect to V . 3. The identity < V [,V > (x, y) = g V (x, y),V (x, y) does not hold here. 1 2 x 1 2 4. The identity (V [)] = V does not hold. 5. Even though the sharp and flat we define here do not satisfy all properties as in local calculus they are good enough in the sense that they guarantee the relations between operators in vector calculus as shown in following sections. 6. Here we use a fixed skew-symmetric vector field V to define sharp and flat. 0 Thus the dual of a 1-form is a symmetric vector field. We can also start with a fixed symmetric vector field V and define the dual of a 0 1-form as a skew-symmetric vector field using the same formulas above for sharp n and flat. We obtain all the same properties listed. For example in R with the trivial NL trivialization one can take V (x, y) = (x + y)/||x + y|| if x 6= −y (and 0 V (x, −x) = 0). 0

3.4 NL Vector Calculus

In this section we will define all the usual operators in vector calculus using the invariant formulations as in local vector calculus. That is vector calculus operators are defined using only operators in exterior calculus such as sharp, flat, 32 exterior derivative, etc (in local calculus this means vector calculus operators can be written in coordinate-free notation). But first of all similar to Section 3.3 we would like to start with general definitions.

Definition 3.4.1 i) The NL gradient ∇f of a 0-form f is a NL vector field defined as

Z 0 0 0 0 0 0 0 0 0 ∇f(x, y) = σ (x, y, x , y , z ) f(z ) λ (V (x , y )) dx dy dz . g x0x 0g M3

Here σ : M5 → is a given kernel, V is a given NL vector field. g R 0g ii) The NL divergence div(V ) of a NL vector field V is a 0-form defined by

Z 0 0 00 00 0 0 00 00 0 0 00 00 div(V )(x) = σ (x, x , y , x , y ) g V (x , y ), λ (V (x , y )) dx dy dx dy . d x0 x00x0 0d M5

Here σ : M5 → is a given kernel, V is a given NL vector ﬁeld. d R 0d iii) The NL Laplacian ∆f of a 0-form f is a 0-form deﬁned by

Z 0 0 0 0 0 0 0 ∆f(x) = σ (x, x , y , z ) f(z ) dx dy dz L M3

Here σ : M4 → is a given kernel. L R

Remark 3.4.2 In (i) since ∇f(x, y) ∈ T we need to use the NL trivialization so x that λ (V (x0, y0)) ∈ T . Similar reason for using NL trivialization in (ii). x0x 0g x

Just as in standard context, we would like to preserve similar relationships among operators in vector calculus. For example gradient and divergence are the adjoint of each other with respect to the inner product on Ω0 (M) and an inner NL product on the space of NL vector ﬁelds which is induced from the inner product on 33

Ω1 (M). Also the Laplacian should be written in terms of gradient and divergence NL and there should be a “divergence theorem”. We now show that the operators in vector calculus which are induced by the NL exterior calculus have special forms, which give us many standard relationships just as in local case.

Theorem 3.4.3 Suppose that the differential D and codifferential operators D∗ correspond to the special operator T in Section 2.3 and the special inner product in Section 2.4. Also suppose that the sharp and flat have special forms as in Definition 3.3.3. i) If σ (x, y, x0, y0, z0) = (K(y, z0) − K(x, z0)) δ (x0) δ (y0) and V = V then g x y 0g 0 the gradient is induced by the exterior calculus, that is ∇f = (Df)]. Here again δ is the Dirac delta function. ii) If σ (x, x0, y0, x00, y00) = 2K(x, x0) W (x0, y0) δ (x00) δ (y00) and V = V d 1 x0 y0 0d 0 then the divergence is induced by the exterior calculus, that is div(V ) = −D∗V [. iii) If σ (x, x0, y0, z0) = −2K(x, x0)(K(y0, z0) − K(x0, z0))W (x0, y0) then the L 1 Laplacian is induced by the exterior calculus, that is ∆f = (D∗D + DD∗)f (the Laplace-Beltrami operator applies on a 0-form).

Proof. i)

] Z 0 0 0 0 0 0 (Df) (x, y) = K(x, x )K(y, y )(f(y ) − f(x )) dx dy V (x, y) 0 M2

Z 0 0 0 0 = (K(y, z ) − K(x, z ))f(z ) dz V (x, y) 0 M so ∇f = (Df)] if σ (x, y, x0, y0, z0) = (K(y, z0) − K(x, z0)) δ (x0) δ (y0) and g x y V = V . 0g 0 34

ii)

∗ [ Z 0 0 0 0 0 0 0 0 0 −(D V )(x) = 2 K(x, x ) W (x , y ) g V (x , y ),V (x , y ) dx dy 1 x0 0 M2

(use the formula of D∗ in Proposition 2.4.5 and notice that we choose W ≡ 1). 0 Thus div(V ) = −D∗V [ if σ (x, x0, y0, x00, y00) = 2K(x, x0) W (x0, y0) δ (x00) δ (y00) d 1 x0 y0 and V = V . 0d 0 iii) (D∗Df + DD∗f)(x) = (D∗D f)(x) = 0 0

Z 0 0 0 0 0 0 0 0 0 0 0 −2 K(x, x )(K(y , z ) − K(x , z ))W (x , y )f(z ) dx dy dz 1 M3

and thus ∆f = (D∗D + DD∗)f if σ (x, x0, y0, z0) = −2K(x, x0)(K(y0, z0) − L K(x0, z0))W (x0, y0). 1

Remark 3.4.4 Since V is skew-symmetric, the gradient is a symmetric vector 0 field and the divergence acts on symmetric vector fields. This is to match up with existing work on NL vector calculus which we will discuss further in the next subsection. As we mention in remark 6 of Remark 3.3.5, we can also choose V to be a 0 symmetric vector field. We then obtain the same formulas for these operators but now the gradient is a skew-symmetric vector field and the divergence acts on skew-symmetric vector fields.

3 Definition 3.4.5 In R the curl of a symmetric vector field is a symmetric vector field defined by [ ] curl(V ) = (∗DV ) 35

Here the Hodge operator (*) is deﬁned such that the identity D∗∗ = (−1)p ∗ D holds.

We now list some familiar properties of these operators as mentioned above.

Theorem 3.4.6 The gradient (∇) and divergence (-div) are the adjoint of each other: (∇f, V ) = (f, −div(V )) for any 0-form f and any symmetric two-point vector ﬁeld V. Here (.,.) on the left hand side is the inner product of two symmetric two-point vector ﬁelds given by

Z Z (V ,V ) = g (V (x, y),V (x, y)) W (x, y) dxdy 1 2 x 1 2 1 M M and (.,.) on the right hand is the inner product of two 0-forms deﬁned in Section 2.4 (since W ≡ 1, this is the standard L2- product of two functions on M). 0

Proof.

Z 0 0 0 0 (∇f, V ) = (K(y, z ) − K(x, z ))f(z ) g (V (x, y),V (x, y)) W (x, y) dxdydz x 0 1 M3

Z 0 0 0 0 0 0 0 0 0 0 = (K(y , x) − K(x , x))f(x) g (V (x , y ),V (x , y )) W (x , y ) dxdx dy x0 0 1 M3 Z 0 0 0 0 0 0 0 0 0 = f(x)K(x, y ) g (V (x , y ),V (x , y )) W (x , y ) dxdx dy x0 0 1 M3 Z 0 0 0 0 0 0 0 0 0 − f(x)K(x, x ) g (V (x , y ),V (x , y )) W (x , y ) dxdx dy x0 0 1 M3 Z 0 0 0 0 0 0 0 0 0 = −2 f(x)K(x, x ) g (V (x , y ),V (x , y )) W (x , y ) dxdx dy = (f, −div(V )) x0 0 1 M3

(Notice that g (V (x0, y0),V (x0, y0)) is a 1-form). x0 0 36

Remark 3.4.7 In [11] the authors distinguish point-based operators and two- point operators. They define three point-based operators first (grad, div, curl), then defined two-point operators as the adjoints of these operators. Here we do not distinguish between point-based and two-point operators thus we only have three operators as in local calculus.

Theorem 3.4.8 “Divergence Theorem”:

Z div(V )(x) dx = 0 M

Proof. Notice that R K(x, x0) dx = 1 and W (x0, y0) g V (x0, y0),V (x0, y0) M 1 x0 0 is a skew-symmetric function of x’ & y’ (1-form), so its integral on M2 is 0.

3 Theorem 3.4.9 In R : i) curl ◦ grad = 0 ii) div ◦ curl = 0

Proof. i) curl(∇f) = (∗D(∇f)[)] = (∗D2f)] = 0 ii) div(curlV ) = −D∗(curlV )[ = −D∗(∗DV [) = −∗D2V [ = 0 (notice that we use the identity D∗∗ = (−1)p ∗ D )

Theorem 3.4.10 The Laplacian can be obtained from the formula

∆f = −div(∇f)

Proof. By using Theorem 3.4.3 and Lemma 3.3.4 we have −div(∇f) = D∗((Df)])[ = (D∗D)f = ∆f.

Remark 3.4.11 One obtains the same formula for the NL Laplacian as in local n ∗ ∗ calculus (note that in local calculus in R the deﬁnition ∆ = DD + D D will 37 lead to n 2 X ∂ f ∆f = − 2 i=1 ∂x i which is equal to −div(∇f) .

Theorem 3.4.12 The Laplace-Beltrami operator is self-adjoint:

(∆ω, η) = (ω, ∆η) and it is positive semideﬁnite:

∗ ∗ (∆ω, ω) = (Dω, Dω) + (D ω, D ω) ≥ 0

(for ω, η ∈ Ωp (M)) NL

3.5 Some special cases of NL vector calculus operators

In [14] the authors develop a nonlocal vector calculus framework which then is applied in image processing. In [11] the authors also develop a nonlocal vector calculus framework which then is applied to linear peridynamic materials. In this section we point out that for some special case of the kernel K we recover some basic operators defined in these articles. Moreover if we choose other forms of the kernel we can get other forms of these operators. 1. About the gradient If K is the Dirac delta function then ∇f(x, y) = (f(y) − f(x))V (x, y). 0 This operator is defined in [11]. It is also similar to the gradient defined in [14], however in [14] it is a scalar operator. On the other hand there are cases where gradient is an integral operator. For 38

2 ||x−y|| n 1 − example for M = R , if we choose K(x, y) = n e 4δ then (4πδ) 2

0 2 0 2 ||x −x|| ||y −y|| 1 Z − − 0 0 0 0 ∇f(x, y) = e 4δ e 4δ (f(y ) − f(x )) dx dy V (x, y) (4πδ)n n 2 0 (R )

2. About the divergence n If M is a domain in R with the standard inner product and if we choose K to be the Dirac delta function then

Z div(V )(x) = 2 V (x, y).V (x, y) W (x, y) dy 0 1 M

This operator is also defined in [11] (slightly different). We can also use different kernel K to get different forms of divergence. For example, 2 ||x−y|| n 1 − in R with the standard inner product and K(x, y) = n e 4δ , we obtain (4πδ) 2

0 2 ||x −x|| 2 Z − 0 0 0 0 0 0 0 0 div(V )(x) = e 4δ V (x , y ).V (x , y ) W (x , y ) dx dy n n 2 0 1 (4πδ) 2 (R )

3. About the Laplacian In the case the kernel K is the Dirac delta function the Laplacian defined here coincides with the definition in [14] (differs by a constant) or in [3]. That is

Z ∆f(x) = −2 (f(y) − f(x) W (x, y) dy (3.2) 1 M n We refer to [14] (p. 1007-1008) for discussion about the Laplacian (3.2) in R . In [11] it is proved that for a domain in n, if one choose W to be (1/2) of the local R 1 Laplacian of the Dirac delta function then the NL Laplacian (3.2) is indeed the 39 local one when applying on smooth functions with compact support. In [4] it is proved that if one choose W to be the heat kernel in n: W (x, y) = 1 R 1 2 ||x−y|| −1 −n/2 − (2δ) (4πδ) e 4δ then for a smooth function f, the NL Laplacian (3.2) converges pointwise to the usual (local) Laplacian of f as δ → 0. The parameter δ is usually referred as the horizon parameter([11]). Notice that the weight function W satisﬁes the normalized condition (2.8). 1

n s Remark 3.5.1 In R , given s ∈ (0, 1), the fractional Laplacian (−4) is deﬁned by s Z f(x) − f(y) (−4) f(x) = C(n, s) × Principal Value of dy n n+2s R ||x − y|| Here C(n, s) is a given constant (see [25]). Thus the fractional Laplacian can be considered as a special case of the NL Lapla- cian (3.2) with a suitable weight function W . However this weight function does 1 not satisfy the condition (2.8).

We can choose diﬀerent forms of the kernel K to get diﬀerent forms of the 2 ||x−y|| n 1 − Laplacian, for example on R with K(x, y) = n e 4δ , we obtain (4πδ) 2 ∆f(x) =

0 2 0 0 2 0 0 2 ||x−x || ||y −z || ||x −z || 2 Z − − − 0 0 0 0 0 0 − e 4δ e 4δ − e 4δ W (x , y )f(z ) dx dy dz (4πδ)n n 3 1 (R ) (3.3) We now show that the NL Laplacian (3.3) can be use to approximate the local Laplacian. We consider the case M = and W is chosen as following R 1

  3 0  03 if |x − y| ≤ δ 0 W (x, y) = 2δ (δ > 0 is a parameter) 1  0 otherwise 40

Again note that this weight function satisﬁes the condition (2.8). Then the NL Laplacian (3.3) becomes ∆f(x) =

0 2 0 0 2 0 0 2 |x−x | |y −z | |x −z | 3 Z − − − 0 0 0 0 − e 4δ e 4δ −e 4δ f(z ) dx dy dz 03 0 0 0 0 0 0 4πδδ x ,y ,z ∈R, |x −y |≤δ (3.4)

Theorem 3.5.2 Suppose that f is an analytic function on R. Then the NL Lapla- cian (3.4) converges pointwise to the local Laplacian of f (that is −f00) as δ, δ0 → 0.

Proof. By change of variable y0 = x0 + t and integrate with respect to x0 ﬁrst, using a well known result on Gaussian integrals, we reduce (3.4) to

0 2 0 2 (z −x−t) (z −x) 3 Z − − 0 0 ∆f(x) = − √ e 8δ − e 8δ f(z ) dz dt (3.5) 03 0 0 2 2πδ δ z ∈R, |t|≤δ

Now we consider the function f of the form f(z0) = (z0 − x)k for a given x (k is an integer). By change of variable z = z0 − x (3.5) becomes

2 (z−t) 2 3 Z − −z k ∆f(x) = − √ e 8δ − e 8δ z dzdt 03 0 2 2πδ δ z∈R, |t|≤δ

2 3 Z k k −z = − √ (z + t) − z e 8δ dzdt (3.6) 03 0 2 2πδ δ z∈R, |t|≤δ It is easy to see that when k = 0, 1 the integral (3.6) is 0 and when k = 2 the integral is -2, that is −f00 (for any δ, δ0 > 0). When k > 2 it can be checked that the integral (3.6) is 0 if k is odd and is a polynomial of two variables δ and δ0 with the constant coeﬃcient equal to 0 if k is 41 even. In fact the integral of the form

2 Z ` ` −z t 1z 2e 8δ dzdt 0 z∈R, |t|≤δ

0` +1 √ ` +1 is 0 if either ` or ` is odd and is proportional to δ 1 ( δ) 2 if both ` and 1 2 1 ` are even. 2 Thus the integral (3.6) converges to 0 as δ, δ0 → 0.

Now for an analytic function f on R, given a point x we can write f as

∞ (k) 0 X f (x) 0 k f(z ) = (z − x) k! k=0 hence the integral (3.5) converges to −f00(x) as δ, δ0 → 0.

Remark 3.5.3 1. In order to obtain the convergence in Theorem 3.5.2 it is necessary to let both parameters δ and δ0 go to 0. Thus we can not choose the weight function W to be a constant (≡ 1). Of course one can choose δ = δ0. 1 2. One can prove the convergence in Theorem 3.5.2 by just assuming that 3 f ∈ C (R)(as in [30]). Our purpose here is to show a simple application of the framework which provides a diﬀerent way beyond any well known results to approximate the local Laplacian using integral operators. 2 |x−y| − 3. It can be veriﬁed that if we choose W (x, y) = √ 1 e 4δ0 we get the 1 4 πδ0 δ0 same result as in Theorem 3.5.2. When M = n we just need to adjust W R 1 accordingly.

4. About the curl 3 If M is a bounded domain in R we can choose all the weight functions to be 1. Then when the kernel K is the Dirac delta function in order for the Lemma 2.6.1 42 to hold, the kernels K of the Hodge operator can be chosen as K = −K = p 1 2 −2K , K = −K and K is a skew-symmetric function of ﬁve variables. Then 0 3 0 0 the curl has an explicit form:

Z 0 00 000 0 00 0 00 0 00 000 curl(V )(x, y) = 3 K (x, y, u , u , u )(V (u , u ).V (u , u )) du du du V (x, y) 0 0 0 M3 43

Chapter 4

Relationships to Local Geometry, Classical Vector Calculus and Discrete Vector Calculus

In this chapter we show the relationships between our NL exterior calculus and the local and the discrete ones. We will show that a NL trivialization may or may not induce the Levi-Civita connection on M and nonlocal operators in vector calculus can converge to their local counterparts. We also point out the relationship between the nonlocal Laplacian and the discrete one.

4.1 Relationship between NL geometry and local geometry

Consider a Riemannian manifold M together with its Levi-Civita connection. One can see that for any two points x, y in M, the parallel translation along a given non-closed curve connecting x and y satisfies all properties of a nonlocal trivialization defined above (if x = y then parallel translation along a closed curve at x may not be the identity map). It is also well known that the Levi-Civita connection can be recovered from parallel translation (in the limit) as following (see [33], Chapter 6): Given two smooth vector fields X, Y , for each point m ∈ M, let c : J → M be a curve with c(0) = m and c0(0) = X ∈ T (J is an interval in containing 0) . m m R Then the covariant derivative of the vector field Y with respect to the vector field X (the Levi-Civita connection) at the point m is given by

τ−1Y − Y (0) c(t) (∇ Y )(m) = lim t X t→0 t 44

Here τ is the parallel translation along the curve c from m = c(0) to c(t). We t note that this limit does not depend on how the curve c is chosen provided that c(0) = m and c0(0) = X , so c can be chosen as a geodesic curve. m So for a NL trivialization λ one can consider the limit as the one above but replace the parallel translation by the NL trivialization:

λ Y − Y (0) c(t)c(0) c(t) lim (4.1) t→0 t

This limit may or may not exist (for any vector ﬁelds X, Y ). If it exists one can recover some properties of the Levi-Civita connection on M (the smoothness of λ may be required). One can also consider the limit (4.1) without the property (iii) of λ (in Chapter 3) and if this limit exists one may recover some properties of an aﬃne connection. It is well known that for each point m ∈ M there exists a neighborhood of m such that m can connect to any point in that neighborhood through a unique minimal geodesic curve. So if the NL trivialization λ is the parallel translation along these geodesic curves (when restrict on that neighborhood) then the limit (4.1) does exist and one recovers the Levi-Civita connection (to recover local geometry we only need to know the NL trivialization locally). We now consider some examples:

n Example 4.1.1 1. The trivial NL trivialization in R is indeed the parallel trans- n lation corresponding to the standard Levi-Civita connection in R so the limit (4.1) does exist and it gives back the Levi-Civita trivialization. 2. If for any two points x, y ∈ n with x 6= y, λ is either a given reflection R xy (not the identity map, for example the reflection given by the matrix diag(−1 1) 2 in R ) or the reflection in the hyperplane through the origin, orthogonal to the 45 vector (x − y) then one can verify that the limit (4.1) does not exist. 2 3. Consider the sphere S with two NL trivializations given above by (3.1). It can be verified that for the one corresponding to the vector (x + y), the limit (4.1) exists and defines the following trivialization:

(∇ Y )(m) = (dY ) (X ) + < X ,Y > m X m m m m

2 2 (for any point m ∈ S and any vector ﬁelds X, Y on S , <, > is the Euclidean 3 3 inner product in R , d is the usual diﬀerential in R ). 2 This trivialization is indeed the Levi-Civita connection on S . One can verify that in this case λ is in fact the parallel translation along the unique minimal geodesic xy curve connecting x and y (an arc of a great circle) except when x, y are antipodal points. On the other hand for the NL trivialization λ corresponding to the vector (x − y) the limit (4.1) does not exist.

Remark 4.1.2 From these examples one may come up with the conclusion that nonlocal Riemannian geometries are more ubiquitous than (local) Riemannian geometries.

4.2 Relationship between NL Vector Calculus and the Local One

In Chapter 3 we already show the convergence between some version of the NL Laplacian and the smooth one in existing work and our new result. n In [11] the authors deﬁne NL partial derivatives in R based on the NL gradient and they prove that under some smoothness conditions, these NL partial derivatives converge to the usual partial derivatives. In [16] the authors present a deep study about NL boundary-value problems with 46 n respect to the NL Laplacian operator in R . They also prove that under some smoothness conditions, these NL boundary-value problems converge to classical boundary-value problems.

4.3 Relationship between NL Laplacian and discrete Laplacian

3 Now suppose that S is a compact smooth surface in R which is discretized by a mesh K with the set of vertices V lying on S. In [5] the authors deﬁne K as an (, η)- approximation of S (, η > 0) if the following conditions hold: i) For a face t ∈ K, its diameter is at most ρ, where ρ is the reach of S (see [5] for the deﬁnition of the reach of a surface). ii) For a face t ∈ K and a vertex p ∈ t, the angle between n - the unit outward p normal vector of S at p and n - the unit outward normal of the plane passing t through t, (n , n ), is at most η. ∠ p t The authors then propose a discrete Laplace operator as following:

2 ||y−x|| h 1 X area(t) X − (L f)(x) = e 4h (f(y) − f(x)) (4.2) K 4πh2 #t t∈K y∈V (t)

Here f is a function : V → R, V (t) is the set of vertices of the face t, #t is the number of vertices of t, h is a positive parameter, area(t) is the area of the face t, || || is the Euclidean norm. They also introduce the functional Laplacian as following

2 ||y−x|| h 1 Z − F f(x) = e 4h (f(y) − f(x)) dy 2 S 4πh S 47

It is then proved in [5] that under some conditions ||Lh f − F hf|| → 0 as L K S ∞ , η → 0 and ||F hf − 4 f|| → 0 as h → 0. Here 4 is the smooth (local) S L S S ∞ Laplace operator on the surface S. Thus the discrete Laplacian Lh converges to K the local Laplacian in the L sense. ∞ Now one can see that the functional Laplacian F h is in fact the NL Laplacian (3.2) S with the weight function (we consider the case when the diﬀerential operator is the discrete one) 2 ||x−y|| 1 − W (x, y) = e 4h 1 8πh2

(the only diﬀerence is the negative sign). This weight function is normalized in the sense of the identity (2.8) in Section 2.4:

Z 2 ||z|| W (||z||) dz = 2 2 1 R

Here W is considered as a function of the distance: W (x, y) = W (||x − y||) (see 1 1 1 [14], p. 6-7 for more details). The parameter h again is referred to as the horizon parameter ([11]). Thus the NL Laplacian can play a role as an intermediate object between the discrete Laplace operator and the smooth one. By using some results from [5] we now propose a discretization of the NL Laplacian with a general weight function W which then converges to the NL Laplacian in 1 the L sense (under some condition). ∞

3 Theorem 4.3.1 Let S be a smooth compact surface in R and K be an (, η) - approximation of S. 48

Deﬁne the discrete operator as following: for a function f : V → R,

X area(t) X (L f)(x) = −2 W (x, y)(f(y) − f(x)) K #t 1 t∈K y∈V (t)

Now assume that W is diﬀerentiable with respect to one variable (it is symmetric) 1 and W (x, y) and || ∂ W (x, y)|| are upper bounded by some constant C (for all 1 ∂y 1 x, y ∈ S). Then for any function f ∈ C1(S), ||L f − 4f|| → 0 as , η → 0 K L ∞ Proof. In [5] the authors prove the following inequality

Z X area(t) X 2 g(y)dy − g(y) ≤ 3(ρL + ||g|| (2 + η) )Area(S) #t ∞ S t∈K y∈V (t) here g : S → R is a Lipschitz function with the Lipschitz constant L. Now let g(y) = W (x, y)(f(y) − f(x)). In order to bound 1

Z X area(t) X |L f(x)−4f(x)| = 2 W (x, y)(f(y)−f(x))dy − W (x, y)(f(y)−f(x)) K 1 #t 1 S t∈K y∈V (t) it is suﬃcient to bound ||g|| and the Lipschitz constant of g. ∞ It is easy to see that ||g|| ≤ 2C||f|| and since f ∈ C1(S) and S is compact ∞ ∞ ||f|| < ∞. Thus ||g|| is bounded. ∞ ∞ The Lipschitz constant of g is upper bounded by ||g0|| = sup ||∇ g(y)||, ∞ y∈S S here ∇ is the (smooth) surface gradient. S One can see that

||∇ g(y)|| = ||(f(y) − f(x))∇ W (x, y) + W (x, y)∇ f(y)|| S S 1 1 S

∂ 0 ≤ 2||f|| || W (x, y)|| + C||∇ f|| ≤ 2C||f|| + C||f || ∞ ∂y 1 S ∞ ∞ ∞ 49

∂ Here ||∇ W (x, y)|| ≤ ||∇ 3W (x, y)|| = || W (x, y)|| ≤ C S 1 R 1 ∂y 1 and since f ∈ C1(S), S is compact, ||f0|| < ∞. ∞ Thus ||L f − 4f|| → 0 as , η → 0. K ∞ 50

Chapter 5

NL Hodge Theory

In this chapter we would like to obtain similar results for the NL Hodge theory as the results for local exterior calculus presented in [2]. The standard and main results we would like to obtain here are Poincaréinequality and Hodge decomposition. We also would like to extend partly results for the Hodge theory on metric spaces presented in [3]. As mentioned in the introduction chapter (Section 1.4), in our work we embed the horizon parameter in the weight functions and fix the Hilbert complex (the L2 de Rham complex). Thus we only have one Hodge theory. In case the exterior derivative is bounded we have the L2-Hodge theory with similar results as in [3]. On the other hand when this operator is unbounded but still ”good enough” (that is it is densely-defined and closed) just as the smooth one, we can also obtain similar results.

5.1 Hodge Theory for the Discrete Derivative

First of all we consider the case when the NL diﬀerential operator is the discrete one d.

Theorem 5.1.1 If the weight functions W (p = 0,1,...) are integrable with respect p to the last variable (and thus to any variable) and bounded in the sense that

1 Z W (v , ..., v , v ) dv ≤ C (5.1) W (v , ..., v ) p+1 0 p p+1 p+1 p p 0 p M 51 almost everywhere for some constant C (depending on the weight functions only) p and for all index p then the operator d : L2(Ωp (M)) → L2(Ωp+1(M)) is a bounded linear map. NL NL Proof. From the deﬁnition of d and the L2 inner product, for a form ω ∈ L2(Ωp (M)) we have NL

p+1 2 X Z 2 ||dω|| = |ω(v , ..., vˆ , ..., v )| W (v , ..., v ) dv ...dv p+2 0 i p+1 p+1 0 p+1 0 p+1 i=0 M

X Z +2 ω(v , ..., vˆ , ..., v ) ω(v , ..., vˆ , ..., v ) W (v , ..., v ) dv ...dv p+2 0 i p+1 0 j p+1 p+1 0 p+1 0 p+1 0≤i

2 Z 2 ||dω|| = (p + 2) |ω(v , ..., v )| W (v , ..., v ) dv ...dv 0 p p+1 0 p+1 0 p+1 Mp+2

p+1 Z +(−1) (p+1)(p+2) ω(v , ..., v ) ω(v , ..., v ) W (v , ..., v ) dv ...dv 0 p 1 p+1 p+1 0 p+1 0 p+1 Mp+2 By the Cauchy - Schwarz inequality,

Z | ω(v , ..., v ) ω(v , ..., v ) W (v , ..., v ) dv ...dv | 0 p 1 p+1 p+1 0 p+1 0 p+1 Mp+2

Z 2 ≤ |ω(v , ..., v )| W (v , ..., v ) dv ...dv 0 p p+1 0 p+1 0 p+1 Mp+2 And by the assumption,

Z 2 |ω(v , ..., v )| W (v , ..., v ) dv ...dv 0 p p+1 0 p+1 0 p+1 Mp+2

Z 2 2 ≤ C |ω(v , ..., v )| W (v , ..., v ) dv ...dv = C ||ω|| p 0 p p 0 p 0 p p Mp+1 52

Thus the operator d is bounded.

Remark 5.1.2 With the assumption 5.1, due to the Theorem above we conclude that the operator d is deﬁned everywhere and closed (its graph is closed). Thus we have the following Hilbert complex

2 0 d 2 1 d 0 → L (Ω (M)) → L (Ω (M)) → ... NL NL

Its dual complex is

d∗ 2 p d∗ 2 p−1 d∗ d∗ 2 0 ... → L (Ω (M)) → L (Ω (M)) → ... → L (Ω (M)) → 0 NL NL NL

(see [2] for more details about Hilbert complexes).

We now give some examples of weight functions satisfying the condition of Theorem 5.1.1. The ﬁrst example is when M has ﬁnite volume, W has the 1 Gaussian form 2 d(x,y) − W (x, y) = C(δ)e δ and W (v , ..., v ) = Q W (v , v ). Obviously 1 p 0 p 0≤i

p Z Y W (v , v ) dv ≤ C(p) 1 i p+1 p+1 M i=0 which is true since W is bounded and M has finite volume. Even in the case that 1 n M = R (thus M does not have finite volume) the condition of Theorem 5.1.1 still holds. n Another example is following: suppose that M is a bounded domain in R . Consider the following weight function 53   C(δ)  s if ||x − y|| ≤ δ W (x, y) = ||x−y|| (s ∈ [0, n − 1]) 1  0 otherwise Also define s Y W (v , ..., v ) = W (v , v ) p 0 p p 1 i j 0≤i 1. For any x ∈ M,

Z Z 1 W (x, y) dy ≤ C(δ) dy (5.2) 1 s M B (δ) ||y − x|| x Here B (δ) is the ball in n with center x and radius δ. By using the spherical x R n coordinates in R with center at x, we can see that the integral on the right hand side of (6.2) is a constant depending on δ only (independent of x). For the inequality (5.1) when p > 1 notice that if W = 0 then W = 0. When p p+1 W > 0 the left hand side of (5.1) is p

v Z u p 1 p+1uY q t W (v , v ) dv . p(p+1) Q W (v , v ) 1 i p+1 p+1 0≤i

Notice that since W (v , ..., v ) > 0,W (v , v ) > 0, thus 1/W (v , v ) is (upper) p 0 p 1 i j 1 i j s q bounded by δ /C(δ) (for any 0 ≤ i < j ≤ p). This means 1/ p(p+1) Q W (v , v ) 0≤i

v u p p uY 1 X tp A ≤ A i p i i=1 i=1 54 q (for non-negative numbers A , ..., A ), one can see that R p+1 Qp W (v , v ) dv 1 p M i=0 1 i p+1 p+1 is again bounded by the integral on the right hand side of (6.2) (multiply with a constant).

Theorem 5.1.3 Suppose that M has ﬁnite volume, all weight functions are positive almost everywhere and for each index p, there exists a positive constant C0 p such that the following condition

1 Z 0 W (v , ..., v , v ) dv ≥ C W (v , ..., v ) p+1 0 p p+1 p+1 p 0 p M p holds almost everywhere. If the codiﬀerential operator d∗ is closed then it has closed range. In fact, Im d∗ = Ker d∗ for p > 0 and Im d∗ = {1}⊥ - the p p−1 0 orthogonal complement of the constants in L2(Ω0 (M)). NL Proof. Due to Proposition 2.4.5 the codiﬀerential operator has the following form (formally): for ω ∈ Ωp (M), NL

p ∗ (−1) (p + 1) Z (d ω)(v , ..., v ) = W (v , ..., v ) ω(v , ..., v ) dv 0 p−1 W (v , ..., v ) p 0 p 0 p p p−1 p−1 0 p−1 M

Since forms are skew-symmetric and weight functions are symmetric, R (d∗ω)(v )dv = M 0 0 0 0. This means Im d∗ ⊂ {1}⊥. 0 On the other hand, let f be a function in {1}⊥ ⊂ L2(Ω0 (M)), that is R f(v) dv = NL M 0. Deﬁne 1-form ω by

1 ω(v , v ) = (f(v ) − f(v )) 0 1 2vol(M) W (v , v ) 1 0 1 0 1 55

Then d∗ω(v ) = −2 R W (v , v ) ω(v , v ) dv = f(v ) since R f(v ) dv = 0. 0 0 M 1 0 1 0 1 1 0 M 1 1 Also ω ∈ L2(Ω1 (M)). In fact NL

2 2 Z 2 1 Z (f(v ) − f(v )) ||ω|| = (ω W )(v , v ) dv dv = 1 0 dv dv 2 1 0 1 0 1 2 2 W (v , v ) 0 1 M 4(vol(M)) M 1 0 1

1 2 ≤ ||f|| 2C0 vol(M) 1 The last inequality is due to the assumption on W (note that W ≡ 1) and 1 0 (f(v ) − f(v ))2 ≤ 2(f(v )2 + f(v )2) (and again M has ﬁnite volume). 1 0 0 1 When p > 1 because d∗2 = 0 we only need to prove that Ker d∗ ⊂ Im d∗ . p−1 p Given a p-form ω ∈ Kerd∗ , deﬁne a (p+1)-form η by p−1

p+1 1 X i η(v , ..., v ) = (−1) (W ω)(v , ..., vˆ , ..., v ) 0 p+1 (p + 2) vol(M) W (v , ..., v ) p 0 i p+1 p+1 0 p+1 i=0

Then d∗ η = ω. In fact for i = 0, ..., p all the integrals R (W ω)(v , ..., vˆ , ..., v ) dv p M p 0 i p+1 p+1 are 0 because ω ∈ Kerd∗ . Also using similar argument for the case p = 0 above p−1 and the assumption on the weight functions one can verify that η ∈ L2(Ωp+1(M)). NL

Kerd∗ Remark 5.1.4 1. The homology group H (M) = p is trivial for p > 0 p Imd∗ p+1 and one dimensional for p = 0. 2. Even though our definitions here are different from those in [3], in the case that the NL differential operator is the discrete one, we obtain similar results as in [3] (see Section 2). Moreover our weight functions are more general. 56

Theorem 5.1.5 Suppose that M has ﬁnite volume and for each index p there exists positive constants C ,C0 such that the weight functions satisfy the following p p condition

0 1 Z C ≤ W (v , ..., v , v ) dv ≤ C W (v , ..., v ) p+1 0 p p+1 p+1 p p p 0 p M holds almost everywhere. Then the following inequality (Poincar´einequality) holds:

⊥ ||ω|| ≤ C(p)||d ω||, ω ∈ (Kerd ) p p for p = 0, 1, ... Here the constant C(p) depends on p only.

Proof. By Theorem 5.1.1 the operator d is bounded. Thus its adjoint operator d∗ is also bounded. By Theorem 5.1.3 the operator d∗ has closed range. By the closed range theorem the operator d has closed range. Thus the bijective map

⊥ d :(Kerd ) → Imd p p p is an isomorphism by the open mapping theorem. It follows that

⊥ C(p) = inf{||d ω|| : ω ∈ (Kerd ) , ||ω|| = 1} > 0 p p and we obtain the Poincar´einequality above. (See also [2] and [3] for more details).

Theorem 5.1.6 Supposed that all conditions of Theorem 5.1.5 hold. Then we have the Hodge decomposition

2 p ∗ L (Ω (M)) = Imd ⊕ Imd ⊕ Ker4 NL p−1 p p 57

Proof. This follows by the Hodge Lemma proved in [3].

Remark 5.1.7 In the Hodge Lemma in [3], the authors require that the coboundary and boundary operators are bounded. However the same results still holds if the co-chain and chain complexes are Hilbert complexes. That is these two operators are only required to be densely-deﬁned and closed. The Hodge decomposition is claimed in [2] for any closed Hilbert complex (either the coboundary or the boundary operator has closed range). Notice that the smooth (local) exterior derivative is an unbounded operator, yet the Hodge decomposition still holds.

The Mixed Formulation Problem Consider the L2 complex in the Remark 5.1.2. Let V p = Dom(Ωp (M)) be NL the domain of the differential operator d. The mixed formulation of the Laplace- Beltrami operator is defined as the problem of finding ω ∈ V p−1, ω ∈ V p p−1 p and a p-harmonic form η (∆η = 0) satisfying

0 0 0 p−1 (ω , ω ) − (dω , ω ) = 0, ω ∈ V p−1 p−1 p−1 p p−1

0 0 0 0 0 p (dω , ω ) + (dω , dω ) + (ω , η) = (f , ω ), ω ∈ V p−1 p p p p p p p 0 0 (ω , η ) = 0, η is harmonic p

Here f is a given p-form. One can see [2] for more details. p Since the Poincar´einequality holds, it has been proved in [2] that the mixed formulation problem is well-posed.

Remark 5.1.8 If the weight functions are not bounded as in Theorem 5.1.1 the derivative operator d could be unbounded. Recall that this is indeed the case for the local diﬀerential operator when it is considered as an operator between the L2 58 n spaces of p-forms and (p+1)-forms on a bounded domain in R (or on many other manifolds).

We show here one example that the NL diﬀerential operator d is unbounded: take M as the unit interval [0, 1] in and W (x, y) = 1/(x − y)2,W (v , ..., v ) = R 1 p 0 p Q W (v , v ). These weight functions do not satisfy the condition of The- 0≤i

2 k k 2 2 Z (f (x) − f (y)) Z (x − y ) ||d f || = k k dxdy = dxdy = 0 k 2 2 M2 (x − y) M2 (x − y)

Z X i j 2 ( x y ) dxdy 2 [0,1] i+j=k−1 is in the order of O(k). Thus ||d f ||/||f || → ∞ as k → ∞. 0 k k For d : L2(Ω1 (M)) → L2(Ω2 (M)), consider the sequence of 1-forms {ω (x, y) = 1 NL NL k xkyk(y − x), k = 1, 2, ...}. It is easy to see that

2 2 Z ω (x, y) Z 2k 2k ||ω || = k dxdy = x y dxdy k 2 M2 (x − y) [0,1]2 has the order of O( 1 ). On the other hand k2

2 2 Z (ω (y, z) − ω (x, z) + ω (x, y)) ||d ω || = k k k dxdydz = 1 k 2 2 2 M3 (x − y) (y − z) (z − x)

Z X i i i 2 ( x 1y 2z 3) dxdydz 3 [0,1] i +i +i =2k−2 1 2 3 59 has the order of O(k). Thus ||d ω ||/||ω || → ∞ as k → ∞. 1 k k One can ﬁnd similar examples for p = 2, .... Even though the operator d could be unbounded, for some special class of weight functions this operator can still be densely-deﬁned and closed, thus it still forms a Hilbert complex as in Remark 5.1.2.

n Theorem 5.1.9 Supposed that M is the closure of a bounded domain in R and

p(p+1)q Q ρ(||v − v ||) ρ(||x − y||) 2 0≤i

Here ρ : [0, ∞) → [0, ∞] is a function satisfying: i) There exists a constant δ > 0 such that ρ(t) > 0 whenever |t| < δ; ii) There exists a constant C(ρ) > 0 such that

ρ(t ) t 1 ≤ C(ρ) 2 ρ(t ) t 2 1 whenever ρ(t ) > 0; 2 n n iii) ρ(||x||) is integrable on R (||.|| is the Euclidean norm in R ). Then the operator d : L2(Ωp (M)) → L2(Ωp+1(M)) is densely-deﬁned. NL NL Proof. The domain of d is

2 p 2 p+1 Dom(d) = {ω ∈ L (Ω (M)) : dω ∈ L (Ω (M))}. NL NL

First of all we claim that the space of all smooth L2 forms C∞(Ωp (M)) ∩ NL L2(Ωp (M)) is a subspace of Dom(d). For an index p, given ω ∈ C∞(Ωp (M))∩ NL NL L2(Ωp (M)), since ω is skew-symmetric, by using Taylor expansion and the fact NL 60 that M is compact, there exists a constant C such that ω

Y |ω(v , ..., v )| ≤ C ||v − v || 0 p ω i j 0≤i

2 s Y |(dω)(v , ..., v )| W (v , ..., v ) ≤ C (p+1)(p+2) ρ(||v − v ||) 0 p+1 p+1 0 p+1 ω 2 i j 0≤i

Then by using the inequality

v u q q uY 1 X tq A ≤ A i q i i=1 i=1 and the facts that M has ﬁnite volume and ρ(||x||) is integrable one can see that dω ∈ L2(Ωp+1(M)). NL Now we show that Dom(d) is dense in L2(Ωp (M)). When p = 0,L2(Ω0 (M)) NL NL is just the standard L2 space of functions on M(W ≡ 1) and C∞(Ω0 (M)) is 0 NL dense in this L2 space, so Dom(d ) is dense in L2(Ω0 (M)). 0 NL When p = 1, we will show that for a given ω ∈ L2(Ω1 (M)), there exists ω ∈ NL C∞(Ω1 (M)) (for any 0 < < δ) such that ω → ω in L2(Ω1 (M)) as → 0+. NL NL ∞ n We know from standard context (e.g [18]) that there exists a function ϕ ∈ C (R ) c R such that ϕ ≥ 0, n ϕ = 1, supp(ϕ) = {x : ||x|| ≤ 1}. R  ω(x, y) if ||x − y|| > 4 Deﬁneω ˜ (x, y) =  0 otherwise Since ω is skew-symmetric (a 1-form),ω ˜ is also skew-symmetric and converges pointwise to ω as → 0. Moreover if ω is continuous then since the domain is 61 compact, it is uniformly continuous and thusω ˜ converges uniformly to ω as → 0. Now just as in standard context (see [18]) consider the convolution:

Z 0 0 0 0 0 0 ω (x, y) = ω˜ (x − x , y − y ) ϕ(x )ϕ(y ) dx dy =

0 0 −2n Z 0 0 x − x y − y 0 0 ω˜ (x , y ) ϕ( )ϕ( ) dx dy .

Sinceω ˜ is skew-symmetric we can see that ω is also skew-symmetric and thus ω ∈ C∞(Ω1 (M)). We will prove that ω → ω in L2(Ω1 (M)) as → 0. NL  NL  0 0 W (x, y) if ||x − y|| ≤ For any 0 < 0 < δ, deﬁne W (x, y) = 1 1  0 otherwise 0 and ||η||2 = R |η(x, y|2 W (x, y) dxdy for any 1-form η. We prove the following 0 1 inequality ||ω || ≤ 8C(ρ)||ω|| (5.3) 0 20

Apply the Minkowski’s integral inequality, we obtain

Z Z 0 0 0 0 0 0 2 0 1/2 ||ω || 0 = | ω˜ (x − x , y − y ) ϕ(x )ϕ(y ) dx dy | W (x, y) dxdy ≤ 1

Z 0 0 n Z 0 0 2 0 o1/2 0 0 ϕ(x )ϕ(y ) |ω(x−x , y−y )| W (x, y) dxdy dx dy ||x0||, ||y0||≤1 ||(x−x0)−(y−y0)||>4 1

Let x00 = x − x0, y00 = y − y0 and rewrite 0 |ω(x − x0, y − y0)|2 W (x, y) = 1

00 00 00 00 0 0 00 00 00 00 2 ρ(||x − y ||) ρ(||(x − y ) + (x − y )||) ||x − y || 2 |ω(x , y )| × ×( ) ||x00 − y00||2 ρ(||x00 − y00||) ||(x00 − y00) + (x0 − y0)|| 62 for ||x00 − y00|| > 4, ||(x00 + x0) − (y00 + y0)|| = ||(x00 − y00) + (x0 − y0)|| ≤ 0 and ||x0||, ||y0|| ≤ 1. Since ||x00 − y00|| > 4 and ||(x00 − y00) + (x0 − y0)|| ≥ ||x00 − y00|| − ||x0 − y0|| ≥ ||x00 − y00|| − 2 we obtain

||x00 − y00|| ≤ 2. ||(x00 − y00) + (x0 − y0)||

Thus by using the property (ii) of ρ we also obtain the inequality

ρ(||(x00 − y00) + (x0 − y0)||) ≤ 2C(ρ) ρ(||x00 − y00||)

On the other hand ||x00 − y00|| ≤ 2||(x00 − y00) + (x0 − y0)|| ≤ 20. From all inequalities above we obtain the inequality (5.3). Now we write

2 2 Z 2 ||ω − ω|| = ||ω − ω|| + |(ω − ω)(x, y)| W (x, y)dxdy. 0 1 ||x−y||>0

Then ||ω − ω|| ≤ ||ω|| + 8C(ρ)||ω|| 0 0 20 and ||ω|| , ||ω|| → 0 as 0 → 0 by the dominated convergence theorem. In fact 0 20 0 0 ω2W is upper bounded by the integrable function ω2W and ω2W converges 1 1 1 pointwise to 0 as 0 → 0. On the second term above when we ﬁx an 0, the weight C(ρ)ρ(0) function W is upper bounded by the constant and thus it goes to 0 as 1 02 → 0. This follows from the known result for standard L2 spaces (see e.g [18]) and the fact that when ω is continuous,ω ˜ converges uniformly to ω as → 0. We can use exactly the same argument above when p > 1. For example we can 63 modify a 2-form ω starting with   ω(x, y, z) if ||x − y||, ||y − z||, ||x − z|| > 4 ω˜ (x, y, z) =  0 otherwise.

Remark 5.1.10 1. If the weight function W is not integrable (considered as a 1 function of one variable z = x − y) the discrete derivative d may be an unbounded operator (we already show an example in R). 2. In order to obtain the Poincar´einequality we are still working to prove that with all conditions in the Theorem 5.1.9 the discrete derivative d is closed and has closed range. The Poincar´einequality for the case p = 0 is claimed in [22].

5.2 Hodge Theory for the Nonlocal Exterior Derivative

We now show some similar results for the NL diﬀerential operator D. We will be working on the operator D = d◦T where T is again the operator (2.5).

Theorem 5.2.1 If the weight functions W (p = 0,1,...) satisfy the following p condition

p 1 Z Y 0 { K(v , v )} W (v , ..., v ) dv ...dv ≤ C 0 0 i p+1 0 p+1 0 p+1 p W (v , ..., v ) Mp+2 i p 0 p i=0 (5.4) almost everywhere for some constant C (depending on the weight functions and p the kernel K only) and for all index p then the operator D : L2(Ωp (M)) → L2(Ωp+1(M)) is a bounded linear map. NL NL 64

Proof. Since Dω = d(T ω), by replacing ω in the proof of Theorem 5.1.1 with T ω we only need to prove that

Z 2 2 |(T ω)(v , ..., v )| W (v , ..., v ) dv ...dv ≤ C ||ω|| . 0 p p+1 0 p+1 0 p+1 p Mp+2

By applying the Cauchy - Schwarz inequality and note that R K(x, y)dy = 1 we M obtain

p 2 Z Y 0 0 0 2 0 0 |(T ω)(v , ..., v )| ≤ K(v , v ) |ω(v , ..., v )| dv ...dv . 0 p p+1 i M i=0 i 0 p 0 p

Thus from the condition (5.4) we have

Z 2 |(T ω)(v , ..., v )| W (v , ..., v ) dv ...dv 0 p p+1 0 p+1 0 p+1 Mp+2

Z 0 0 2 0 0 0 0 2 ≤ C |ω(v , ..., v )| W (v , ..., v ) dv ...dv = C ||ω|| . p p p Mp+1 0 p 0 p 0 p Remark 5.2.2 When the kernel K is the Dirac delta function Theorem 5.2.1 reduces to Theorem 5.1.1.

Theorem 5.2.3 Suppose that M has ﬁnite volume, all the weight functions are ∗ positive almost everywhere and the restriction operator T | ∗ : KerD → KerD p−1 p−1 KerD∗ is invertible (for all index p > 0 and when p = 0, the operator is p−1 ⊥ ⊥ 0 T | ⊥ : {1} → {1} ). Also suppose that there exists a positive constant C {1} p such that the following condition

1 Z 0 W (v , ..., v , v ) dv ≥ C W (v , ..., v ) p+1 0 p p+1 p+1 p 0 p M p 65 holds almost everywhere. Then Im D∗ = Ker D∗ for p > 0 and Im D∗ = {1}⊥ p p−1 0 - the orthogonal complement of the constants in L2(Ω0 (M)). NL Proof. The proof here is similar to the proof of Theorem 5.1.3. For p = 0 it is clear that Im D∗ ⊂ {1}⊥ (note that R K(x, y)dy = 1). Given 0 M f ∈ {1}⊥ ⊂ L2(Ω0 (M)), construct 1-form ω as following NL

1 −1 −1 ω(v , v ) = (T f(v ) − T f(v )). 0 1 2vol(M) W (v , v ) 1 0 1 0 1

Then D∗ω = f. In fact we ﬁrst look for ω in the following form 0

1 ω(v , v ) = (g(v ) − g(v )), 0 1 2vol(M) W (v , v ) 1 0 1 0 1 where g is a 0-form. Then ∗ Z 0 0 0 0 0 D ω(v ) = −2 K(v , v )(ωW )(v , v ) dv dv = 0 0 0 0 1 0 1 0 1 1 Z 0 0 − g(v ) dv + T g(v ). vol(M) 1 1 0

Since T : {1}⊥ → {1}⊥ is invertible choose T g = f. Then {1}⊥

Z Z Z 0 = f(v ) dv = T g(v ) dv = g(v ) dv 0 0 0 0 0 0 and thus D∗ω = f. 0 Also using the same argument as in the proof of Theorem 5.1.3, we have

2 1 −1 2 ||ω|| ≤ ||T f|| 2C0 vol(M) 1 66 and so ω ∈ L2(Ω1 (M)). NL When p > 0, given a p-form ω ∈ KerD∗ , construct a (p+1)-form η as following p−1

p+1 1 X i −1 η(v , ..., v ) = (−1) (W T ω)(v , ..., vˆ , ..., v ) 0 p+1 (p + 2) vol(M) W (v , ..., v ) p 0 i p+1 p+1 0 p+1 i=0

∗ ∗ ∗ (T | ∗ : KerD → KerD is invertible) Then D η = ω and η ∈ KerD p−1 p−1 p p−1 L2(Ωp+1(M)) (using the same argument as in the proof of Theorem 5.1.3). NL

Theorem 5.2.4 Suppose that all conditions of Theorem 5.2.1 and Theorem 5.2.3 hold. Then the Poincar´einequality holds:

⊥ ||ω|| ≤ C(p)||D ω||, ω ∈ (KerD ) p p for p = 0, 1, ... Here the constant C(p) depends on p only.

Proof. We use the same argument as in the proof of Theorem 5.1.5.

Remark 5.2.5 1. Theorem 5.1.5 is a special case of Theorem 5.2.4. 2. All conditions of Theorem 5.2.1 and Theorem 5.2.3 can be achievable if M is compact, K is the heat kernel on M, W has the Gaussian form W (x, y) = 1 1 C(δ) exp(−d(x, y)2/δ) and W (v , ..., v ) = Q W (v , v ). Note that p 0 p 0≤i

Chapter 6

n Another Model for Nonlocal Exterior Calculus in R

In this chapter we derive a diﬀerent model for nonlocal exterior calculus in n the Euclidean space R based on the model presented in previous chapters, which satisﬁes more intrinsic properties and uses a natural L2 inner product.

6.1 Nonlocal Forms

As mentioned in section 2.1, let e (i = 1, .., n) be the ith unit row vector i n n p n in R . Let ∧ be the usual wedge product in R and ∧ (R ) be the vector space 0 n of all p-multivectors (p = 0, ..., n; ∧ (R ) = R). The set

{e ∧ e ∧ ... ∧ e | 1 ≤ i < i < ... < i ≤ n} i i i 1 2 p 1 2 p is a basis of this vector space.

n n p+1 p n Deﬁnition 6.1.1 A nonlocal p-form on R is a map (R ) → ∧ (R ), which is skew-symmetric. Thus a nonlocal p-form ω can be written as

X ω(v , ..., v ) = ω (v , ..., v ) e ∧ ... ∧ e (6.1) 0 p i ...i 0 p i i 1≤i

where the coeﬃcients ω are skew-symmetric functions ( n)p+1 → i ...i R R 1 p 68

Remark 6.1.2 Each component of a nonlocal p-form here is just a nonlocal p- form deﬁned in Chapter 2. We also denote the vector space of all p-forms as Ωp . NL It is clear that Ωp = {0} if p > n. NL

6.2 Nonlocal Exterior Derivative (D)

In Section 2.4, we deﬁne weight functions W . In n and when p = 1 there p R are two types of the weight function W we are interested in, one is a Gaussian 1 kernel and one is a cut-oﬀ function. If p > 1 we can take

Y W (v , ..., v ) = W (v , v ) p 0 p 1 i j 0≤i

Deﬁnition 6.2.1 The NL exterior derivative D :Ωp → Ωp+1 is a linear NL NL operator deﬁned as following: (Dω)(v , ..., v ) = 0 p+1

X q (d ω )(v , ..., v ) W (v , ..., v ) e ∧e ∧...∧e i ...i 0 p+1 p+1 0 p+1 1 i i 2≤i <...

Remark 6.2.2 1. Here we use vector e in the definition of the operator D, thus 1 all coefficients of ω which contain the index 1 do not appear in the expression of D. In general one can use any constant vector α = Pn α e instead of e , however i=1 i i 1 it is difficult to write down all explicit formulas later on due to the problem of 69 index ordering. 2. It is clear that D2 = 0 since e ∧ e = 0 1 1

6.3 Codiﬀerential Operator (D∗)

Deﬁnition 6.3.1 For ω, η ∈ Ωp , we deﬁne the L2-inner product of these two NL forms as

X Z (ω, η) = ω (v , ..., v ) η (v , ..., v ) dv ...dv n p+1 i ...i 0 p i ...i 0 p 0 p 1≤i

Remark 6.3.2 We note that the weight function does not appear in the L2- product here, so it can be considered as a natural inner product. This is not the case in Section 2.4.

Deﬁnition 6.3.3 The codiﬀerential operator D∗ is the operator which is (formally) adjoint to the exterior derivative operator D with respect to the L2-product above. This means D∗ :Ωp → Ωp−1 is a linear operator such that NL NL

∗ (Dω, η) = (ω, D η) for any (p-1)-form ω and any p-form η.

Similar to Section 2.4 we can write down the explicit formula for the operator D∗ as following: 70

(D∗ω)(v , ..., v ) = (−1)p (p + 1) × 0 p−1

X h Z q i ω (v , ..., v ) W (v , ..., v ) dv e ∧ ... ∧ e (6.4) n 1i ...i 0 p p 0 p p i i 2≤i <...

Here ω ∈ Ωp is written as in (6.1). NL

6.4 NL Laplace-Beltrami Operator (revisit)

As usual the Laplace-Beltrami operator ∆ is a linear operator : Ωp → Ωp NL NL deﬁned by ∆ = DD∗ + D∗D. For a 0-form f ∈ Ω0 we obtain the same formula for ∆f as in section 3.4: NL

Z (∆f)(v ) = −2 (f(v ) − f(v )) W (v , v ) dv 0 n 1 0 1 0 1 1 R

So this is the same as in Section 3.4. n k n k Also if f is a vector function R → R we can deﬁne ∆f: R → R using the same formula.

6.5 Hodge Operator (∗)

Deﬁnition 6.5.1 The Hodge operator is a linear operator ∗ :Ωp → Ωn−p NL NL deﬁned as following: for ω ∈ Ωp which is written as NL

X ω = ω e ∧ ... ∧ e , i ...i i i 1≤i

X j ...j (∗ω)(v , ..., v ) = K 1 n−p(v , ..., v ) × 0 n−p 0 n−p 1≤j <...

Lemma 6.5.2 If all the kernels containing the super index 1 (K1... and K1...) are 1p 2p ﬁxed and all other kernels are chosen as following:

j ...j n n − p + 2 K 1 n−p−1(v , ..., v ) = (−1) × 1 p+1 0 n−p−1 p + 1

Z 1j ...j q K 1 n−p−1(v , ..., v ) W (v , ..., v ) dv n 0 n−p n−p 0 n−p n−p R 1p and

i ...i Z 1i ...i q K 1 p(v , ..., v ) = K 1 p(v , ..., v ) W (v , ..., v ) dv n−p+1 n+1 n n−p n+1 p+1 n−p n+1 n−p 2p R 2 p+1

(2 ≤ j < ... < j ≤ n, 2 ≤ i < ... < i ≤ n, p = 0, ..., n − 1) 1 n−p−1 1 p then the following intrinsic identity holds:

∗ p D ∗ ω = (−1) ∗D ω (6.6) for any ω ∈ Ωp (p = 0, .., n). NL 72

Proof: By some direct calculation using formulas (6.2), (6.4), (6.5) it can be shown that: The left hand side of the identity (6.6) is (D∗∗ ω)(v , ..., v ) = (−1)n−p(n − p + 2) × 0 n−p−1

X h Z 1j ...j q i K 2 n−p(v , ..., v ) W (v , ..., v ) dv × n 0 n−p n−p 0 n−p n−p 2≤j <...

X j ...j (∗D ω)(v , ..., v ) = (p + 1) K 2 n−p(v , ..., v ) × 0 n−p−1 0 n−p−1 2≤j <...

Remark 6.5.3 1. Note that K1...n and K1..n can be considered as n-forms and 10 2n ˆ they can be chosen to be the same. Also for p = 1, all kernels K1...i...n (i = 2, .., n) 11 can be chosen to be the same as K2...n. Similarly for p = n − 1 all kernels 11 ˆ K1...i...n (i = 2, .., n) can be chosen to be the same as K2...n . Then all other 2 n−1 2 n−1 kernels are uniquely determined. So we have the consistency here comparing to the (local) Hodge operator in the sense that in order to deﬁne this operator, one needs to ﬁx a volume form, that is a n-form. Here the natural n-form one can 73 choose is the volume of the n-simplex (or n-tuple): vol([v , ..., v ]) or the sign of 0 n this function (note that in n, vol([v , ..., v ]) is a skew-symmetric function of R 0 n (n+1) variables v , ..., v only). 0 n 2. Due to the nonlocality, the Hodge operator here is not an isometry and does not satisfy the identity ∗∗ ω = (−1)p(n−p) ω (for ω ∈ Ωp ) NL

6.6 Sharp (]) and Flat ([) Operators

Consider the skew-symmetric two-point vector field α : n× n → n, α (x, y) = 1 R R R 1 (y − x)/||y − x|| if x 6= y (this is the vector field V mentioned in section 3.3). We 0 first claim the following result:

Lemma 6.6.1 There exists a set of skew-symmetric two-point vector ﬁelds {α , ..., α } 2 n such that for any x 6= y, {α (x, y), ..., α (x, y)} is an orthonormal basis of n. 1 n R

Proof: For any pair (x, y) ∈ n × n such that x 6= y, let i (x, y) = min{1 ≤ i ≤ R R 0 i i n i th n : y − x 6= 0}. Here for x ∈ R , x is the i component of x. If i = 1, we let 0 β (x, y) = (−(y2−x2), y1−x1, 0, ..., 0); β (x, y) = (−(y3−x3), 0, y1−x1, 0, .., 0); ...; β (x, y) = 2 3 n (−(yn − xn), 0, ..., 0, y1 − x1). That is for any index 2 ≤ k ≤ n, the 1st component of β is −(yk − xk), the kth k component is (y1 − x1), the rest are 0. Similarly if i = 2, that is y1 − x1 = 0 and y2 − x2 6= 0, we let β (x, y) = 0 2 (y2 − x2, 0, ..., 0) and for k = 3, .., n, the 2nd component of β (x, y) is −(yk − xk), k the kth component is (y2 − x2) and the rest are 0 (in fact β does follow this rule: 2 we interchange the ﬁrst and second components). So we can construct β , ..., β using the same rule as above. Now one can see that 2 n if x 6= y, the set {α (x, y), β (x, y), ..., β (x, y)} is indeed linear independent and 1 2 n 74 all vectors β (x, y) are orthogonal to α (x, y). Using the Gram-Schmidt process k 1 we obtain an orthonormal basis {α (x, y), ..., α (x, y)} of n. 1 n R

Remark 6.6.2 1. A set {α , ..., α } with the properties above is not unique. If 1 n n is even, such a set can be constructed in a much simpler way without introducing the index i . For example, in 4, we can take 0 R β (x, y) = y − x; α (x, y) = β (x, y)/||β (x, y)|| 1 1 1 1 β (x, y) = (−(y2−x2), y1−x1, −(y4−x4), y3−x3); α (x, y) = β (x, y)/||β (x, y)|| 2 2 2 2 β (x, y) = (y3−x3, −(y4−x4), −(y1−x1), y2−x2); α (x, y) = β (x, y)/||β (x, y)|| 3 3 3 3 β (x, y) = (y4−x4, y3−x3, −(y2−x2), −(y1−x1)); α (x, y) = β (x, y)/||β (x, y)|| 4 4 4 4 (for any x 6= y). 2. One can also see that these vector fields may not be continuous everywhere in the classical (local) sense. 3. The vector fields α are in fact 1-forms as in Definition 6.1.1. k

Now suppose that such a basis {α } as above is given. Similarly to the i Definition 3.3.3 we define NL sharp and flat operators. Denote the vector space of all symmetric two-point vector fields V such that V (x, x) = 0 (for any x) by V . sym

Definition 6.6.3 i) The sharp operator ] is a linear operator: Ω1 → V NL sym which maps any 1-form ω, ω = P ω e to a vector field ω] ∈ V defined by i i sym ω] = P ω α . i i ii) The flat operator [ is a linear operator: V → Ω1 which maps any vector sym NL field V ∈ V to a 1-form V [ defined by V [ = P (V.α )e , here (.) is the usual sym i i i n Euclidean inner product in R . 75

Remark 6.6.4 It is easy to see that the sharp and ﬂat operators are inverse of each other.

6.7 Vector Calculus (revisit)

Similar to the Section 3.4 now we can deﬁne all basic operators in calculus.

Definition 6.7.1 The NL gradient (∇f) of a 0-form f is a vector field in V sym defined as

] q (∇f)(v , v ) = (Df) (v , v ) = (f(v ) − f(v )) W (v , v ) α (v , v ) 0 1 0 1 1 0 1 0 1 1 0 1

Definition 6.7.2 The NL divergence div(V ) of a vector field V ∈ V is a sym 0-form defined by

∗ [ Z q div(V )(v ) = −(D V )(v ) = 2 V (v , v ).α (v , v ) W (v , v ) dv 0 0 n 0 1 1 0 1 1 0 1 1 R

Remark 6.7.3 Similar to Theorem 3.4.10 one can check that the Laplacian can be expressed as ∆f(x) = −div(∇f)

Definition 6.7.4 In 3 the NL curl operator of a vector field V ∈ V is a R sym vector field in V defined as sym

[ ] curl(V ) = (∗DV ) 76

Remark 6.7.5 By a direct calculation one can obtain curl(V )(v , v ) = 0 1

2 h Z 13 q i K (v , v ) K (v , v , v ) d(X.α )(v , v , v ) W (v , v , v ) dv dv dv α (v , v ) 12 0 1 22 2 3 4 3 2 3 4 2 2 3 4 2 3 4 2 0 1

3 h Z 12 q i +K (v , v ) K (v , v , v ) d(X.α )(v , v , v ) W (v , v , v ) dv dv dv α (v , v ) 12 0 1 22 2 3 4 2 2 3 4 2 2 3 4 2 3 4 3 0 1

Thus curl(V) is perpendicular to α . 1

Similar to the model presented in previous chapters we also obtain the following important relations between grad, div and curl as in classical (local) calculus:

Theorem 6.7.6 i) curl ◦ grad = 0 ii) div ◦ curl = 0

6.8 NL Wedge Product (Λnl)

Deﬁnition 6.8.1 The wedge product of a p-form ω and a q-form η is a (p+q)- form deﬁned as following: Write X ω = ω e ∧ ... ∧ e i ...i i i 1≤i

nl X X ωΛ η = ω ∧ η j ...j nl {i ...i }/{j ...j } 1≤i

e ∧ ... ∧ e , i i 1 p+q here ∧ is the operator deﬁned in Section 2.7. nl

Remark 6.8.2 This wedge operator satisﬁes all properties mentioned in Section 2.7. 78

Chapter 7

Ongoing and Future Works

7.1 Ongoing Works

We would like to find out other classes of the weight functions such that while the NL derivative D may not be bounded, it is still densely-defined and closed and thus (L2(Ωp (M)), D) still forms a Hilbert complex and further more NL the Poincaréinequality and the Hodge decomposition still hold. In [3, 30] the authors show that under some conditions of the manifold M, the scaled cohomology is finite dimensional and isomorphic to the local de Rham cohomology when the scale α is small enough. Even though we are more interested in the computational applications just as in [2], we also would like to find out if the NL de Rham cohomology is finite dimensional and isomorphic to the smooth one in the case the NL exterior derivative D is unbounded (of course under some conditions of the manifold M).

7.2 Future Works

Inspired by the works in [2] on the approximation theory of the (local) de Rham complex, we would like to obtain similar results for our NL de Rham complex. That is we need to find, for each h (a standard approximation parameter), a subcomplex (V , D) of the complex (L2(Ωp (M)), D). This subcomplex has to h NL be finite-dimensional and is a ”good” approximation of the NL de Rham complex. In other words we need to find out at least a good finite element approximation 79 of NL forms. Also we need to find a bounded projection π from the complex h (L2(Ωp (M)), D) to the subcomplex (V , D) . With these tools we can study NL h the stability and convergence of the mixed formulation method, error estimates of the finite element approximation, etc. 80 References

[1] D. Arnold, R. Falk and R. Winther, Finite Element Exterior Calculus, Homo- logical Techniques, and Applications, Acta Numerica, 15:1-155, 2006.

[2] D. Arnold, R. Falk and R. Winther, Finite Element Exterior Calculus: from Hodge Theory to Numerical Stability, AMS, Vol. 47, Number 2, 281-354, 2010.

[3] L. Bartholdi, T. Schick, N. Smale, S. Smale, Hodge Theory on Metric Spaces, 2009, http://arxiv.org/abs/0912.0284.

[4] M. Belkin and P. Niyogi, Towards a Theoretical Foundation for Laplacian- Based Manifold Methods, Journal of Computer and System Sciences, Vol. 74, Issue 8, Pages 1289-1308, 2008.

[5] M. Belkin, J. Sun, Y.Wang, Discrete Laplace Operator on Meshed Surfaces, Annual Symposium on Computational Geometry, Proceedings of the twenty- fourth annual symposium on Computational geometry, http://www.cse.ohio-state.edu/∼yusu/papers/ﬁnalscg laplace.pdf.

[6] Elie´ Cartan, Sur certaines expressions diﬀe´rentielles et le proble`me de Pfaﬀ, 1989.

[7] Fan R. K. Chung, Spectral Graph Theory, Number 92 in CBMS Regional Conference Series in Mathematics, American Mathematical Society, 1997.

[8] R.R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker, Geometric Diffusion as a Tool for Harmonic Analysis and Struc- ture Definition of Data, Part I: Diffusion Maps, Proceedings of the National Academy of Sciences, 102(21):74267431, 2005. 81

[9] K. Cottrill-Shepherd and M. Naber, Fractional Diﬀerential Forms, J. Math. Phys. 42 (2001), 2203-2212.

[10] M. Desbrun, A. Hirani, M. Leok and J. Marsden, Discrete Exterior Calculus, 2003, http://www.math.ucsd.edu/∼mleok/pdf/DeHiLeMa2003 dec.pdf.

[11] Q.Du, M. D. Gunzburger, R. B. Lehoucq, K. Zhou, A Nonlocal Vector Cal- culus, Nonlocal Volume-Constrained Problems, and Nonlocal Balance Laws, 2012, Mathematical Models and Methods in Applied Sciences, World Scien- tiﬁc Publishing Company.

[12] Q. Du and K. Zhou, Mathematical analysis for the peridynamic nonlocal con- tinuum theory, Math. Model. Numer. Anal. 45 (2011), 217234.

[13] Harley Flanders, Diﬀerential Forms with Applications to the Physical Sciences, Dover Publications, 1989.

[14] G. Gilboa and S. Osher, Nonlocal Operators with Applications to Image Pro- cessing, Multiscale Modeling Simulation, 7(3):1005-1028, 2008.

[15] Alexander Grigor’yan, Heat Kernels on Weighted Manifolds and Applications, 2005, http://www.math.uni-bielefeld.de/ grigor/wma.pdf.

[16] M. Gunzburger, R. B. Lehoucq, A Nonlocal Vector Calculus with Appli- cation To Nonlocal Boundary Value Problems, Sandia National Lab, 2010, http://www.sandia.gov/∼rblehou/snl-sand2009-4666J.pdf.

[17] Anil N. Hirani, Discrete Exterior Calculus, PhD thesis, 2003, Cal Tech, http://thesis.library.caltech.edu/1885/3/thesis hirani.pdf.

[18] Lars H¨ormander, Linear Partial Diﬀerential Operators, Springer-Verlag, 1963. 82

[19] J. Jost, Riemannian Geometry and Geometric Analysis, 4th edition, Springer- Verlag, 2005.

[20] V. Katz, Diﬀerential Forms: Cartan to de Rham, Archive for History of Exact Sciences 33, 321-336, 1985.

[21] M. Meerschaert, J. Mortensen and S. Wheatcraft, Fractional Vector Calculus for Fractional Advection-Dispersion, Physica A 367 (2006) 181-190.

[22] Tadele Mengesha, Qiang Du, Analysis of the Peridynamic Model with a Sign Changing Kernel, In preparation, 2012

[23] B. Mohar, The Laplacian Spectrum of Graphs, In Y. Alavi, G. Chartrand, O. R. Oellermann, A. J. Schwenk (Eds.), Graph Theory, Combinatorics, and Applications, Wiley, volume 2, pages 871898, 1991.

[24] B. Nadler, S. Lafon, R.R. Coifman, and I.G. Kevrekidis, Diﬀusion Maps, Spec- tral Clustering, and the Reaction Coordinates of Dynamical Systems, Appl. Comput. Harmon. Anal. 21 (2006), 113127.

[25] Eleonora Di Nezza, Giampiero Palatucci, Enrico Valdinoci, Hitchhiker’s Guide to the Fractional Sobolev Spaces, 2011, http://arxiv.org/abs/1104.4345

[26] M. Patel, Principia Physica, 2000, http://arxiv.org/abs/gr-qc/0002012.

[27] S. Silling, Reformulation of Elasticity Theory for Discontinuities and Long- Range Forces, J. Mech. Phys. Solids 48 (2000), 175209.

[28] S. Silling, M. Epton, O. Weckner, J. Xu, and E. Askari, Peridynamic States and Constitutive Modeling, J. Elast. 88 (2007), 151184.

[29] S. Silling and R. Lehoucq, Peridynamic Theory of Solid Mechanics, Adv. Appl. Mech. 44 (2010), 7369, doi:10.1016/S0065-2156(10)44002-8. 83

[30] Nat Smale, Steve Smale, Abstract and Classical Hodge-De Rham The- ory, 2011, http://www6.cityu.edu.hk/ma/doc/people/smales/abstract-hodge- derham.pdf.

[31] Edwin H. Spanier, Algebraic Topology, Springer-Verlag, New York, 1981.

[32] Michael Spivak, Calculus on Manifolds: A Modern Approach to Classical The- orems of Advanced Calculus, Westview Press, 1965.

[33] Michael Spivak, A Comprehensive Introduction to Diﬀerential Geometry Volume 2, Second Edition, Pulish or Perish Inc., 1979.

[34] V. Tarasov, Fractional Vector Calculus and Fractional Maxwell’s Equations, Annals of Physics, 2008, 323, 2756-2778.

[35] K. Zhou and Q. Du, Mathematical and Numerical Analysis of Linear Peridy- namic Models with Nonlocal Boundary Conditions, SIAM J. Numer. Anal. 48 (2010), 17591780, doi:10.1137/090781267.

[36] KˆosakuYoshida, Functional Analysis, Springer-Verlag, 1980 Vita

Thinh Le was born in Thanh Hoa province, Vietnam in 1979. He is the middle child in his family. Le majored in mathematics at the Hanoi National Uni- versity of Education, Vietnam and earned a bachelor’s degree in 2001 and a master degree in 2003. From 2003 to 2005 he was a lecturer at the Department of Math- ematics, Hanoi National University of Education. In August 2005, he enrolled in the Ph.D. program in Mathematics at the Pennsylvania State University. From August 2005 to July 2007 he did research in Mathematical Physics. Since August 2007 he has studied Applied and Computational Mathematics under the supervi- sion of Professor Qiang Du. His current research interests include Applications of Geometry in Material Science and Nonlocal Mathematical Models. His work at Penn State involves Motion of Microstructures and Nonlocal Calculus.