DOCSLIB.ORG

Conformally Invariant Variational Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conformally Invariant Variational Problems Conformally Invariant Variational Problems. Tristan Rivi`ere1 1Department of Mathematics, ETH Zentrum CH-8093 Z¨urich, Switzerland. 1 I Introduction These lecture notes form the cornerstone between two areas of Mathematics: calculus of variations and conformal invariance theory. Conformal invariance plays a significant role in many areas of Physics, such as conformal field theory, renormalization theory, turbulence, general relativity. Naturally, it also plays an im- portant role in geometry: theory of Riemannian surfaces, Weyl tensors, Q-curvature, Yang-Mills fields, etc... We shall be con- cerned with the study of conformal invariance in analysis. More precisely, we will focus on the study of nonlinear PDEs aris- ing from conformally invariant variational problems (e.g. har- monic maps, prescribed mean curvature surfaces, Yang-Mills equations, amongst others). 2 II Conformal transformations - some funda- mental results. Conformal invariance is a universal property. A transformation is called conformal when it preserves angles infinitesimally, that is, when its differential is a similarity at every point. Unlike in higher dimensions, the group of conformal transformations in two dimensions is very large ; it has infinite dimension. In fact, it contains as many elements as there are holomorphic and an- tiholomorphic maps. This particularly rich feature motivates us to restrict our attention on the two-dimensional case. Although we shall not be concerned with higher dimension, the reader should know that many of the results presented in these notes can be generalized to any dimension. Definition II.1. A C1 map u between two riemannian man- ifolds (M m,g) and (N n, h) is said to be conformal if at every m m point p M dup is a composition of isometries from TpM ∈ m into dup(TpM ) and dilations. This is equivalent to p M m X,Y T M m < du X,du Y > X Y ∀ ∈ ∀ ∈ p p · p · h | |g| |g = dup X h dup Y h <X,Y >g | · | | · | (II.1) Where < , >g and < , >h denotes respectively the scalar · · m · · n ✷ products g on TpM and h on dup(TpM) C Tu(p)N . Lemma II.1. A map u is conformal if and only if there exists a function λ on M m such that p M m X,Y T M m (II.2) ∀ ∈ ∀ ∈ p < du X,du Y > = e2λ(p) <X,Y > p · p · h g λ(p) is called the conformal factor at p. ✷ This result comes from the following Lemma in linear algebra. 3 Lemma II.2. Let A (Rm, Rn) the space of linear maps ∈ L from Rm into Rn Assume that A =0. Then 6 m X,Y R <X,Y > m =0 (II.3) ∀ ∈ R = <AX,AY > n =0 ⇒ R Where <,>Rk denotes the canonical scalar product on the eu- clidian space Rk if and only if there exists λ R such that ∈ m 2λ X,Y R <AX,AY > n = e <X,Y > m (II.4) ∀ ∈ R R Proof of Lemma II.2 We assume II.2 Let X Rm , X =0 We introduce L Rm Rn given by ∈ 6 x −→ 2 m AX y R L Y : = <AX,AY > n | | <X,Y > m ∀ ∈ X R − X 2 R | | m Denote (X)⊥ the subspace of R made of Vectors Y orthgonal to X. We have LXX =0 L (X) =0 X ⊥ m RX (X)⊥ = R ⊕ This clearly implies that L 0 X ≡ For any X and Y in Rm 0 we then have \{ } AX 2 <AX,AY > n = | | <X,Y > m R X 2 R | | (II.4b) AY 2 n m <AX,AY >R = | Y 2| <X,Y >R | | 4 this implies 2 2 m AX AY X,Y R 0 | | <X,Y > m = | | <X,Y > m ∀ ∈ \{ } X 2 R Y 2 R | | | | (II.5) if <X,Y >= 0 then we have 6 AX AY | | = | | (II.6) X Y | | | | if <X,Y >=0 and X = 0 and Y =0 6 6 Let Z = X+Y <X,Z >=0 and <Y,Z >=0 2 6 6 and from (II.5) we deduce AX AZ AY | | = | | = | | X Z Y | | | | | | Hence (II.6) holds for any pair of vectors in Rm 0 and since AX λ \{ } A =0 | X | = e for some λ R 6 | | ∈ This implies (II.4) . It is clear that (II.4) implies (II.3) and we have proved then Lemma II.2 . ✷ Famous conformal transformations are given by the holomorphic or the antiholomorphic maps from a domain C C into C. This characterizes uniquely the conformal transformations from a 2- dimentional domain of C into C. We have indeed the following result. Proposition II.1. Let u be a C1 map from a connected domain of C into C. is conformal if and only if on one has U U U 5 1 either ∂ u = (∂ u i∂ u) 0 (II.7) z 2 x1 − x2 ≡ 1 or ∂ u = (∂ u + i u) 0 (II.8) z¯ 2 x1 x2 ≡ ✷ Proof of Proposition II.1 Because of (II.1) a map u from a 2-dimensional domain of U C into a manifold (N n, h) is conformal if and only if one has <∂x1 u,∂x2 u>h=0 (II.9) ∂ u = ∂ u | x1 |h | x2 |h in the case of U : C C C a direct computation gives U −→ ∂u ∂u¯ 1 = [(∂ u + ∂ u )+ i (∂ u ∂ u )] ∂z ∂z 4 x1 1 x2 2 x1 2 − x2 1 [(∂ u ∂ u ) i (∂ u + ∂ u )] (II.10) x1 1 − x2 2 − x2 1 x1 2 1 = ∂ u 2 ∂ u 2 2i<∂ u,∂ u> 4 | x1 | − | x2 | − x1 x2 where < , > denotes the scalar product in C. Observe that · · the quadratic 1 0 1 0 form Φ(u) given by − ⊗ − ∂u ∂u¯ Φ(u)= dz dz ∂z ∂z ⊗ is invariant under holomorphic change of coordinates: for w(z) satisfying ∂z¯w = 0 one has locally away from zeros of w′ ∂u ∂u¯ φ(u)= dw dw ∂w ∂w ⊗ 6 This form is called the Hopf differential of the mapping u. From (II.10) one sees that u is conformal if and only if ∂zu ∂zu¯ =0 2 which means that either u is antiholomorphic ,∂zu = 0, or u is holomorphic, ∂zu¯ = 0. Which proves proposition II.1. ✷ Thus in two dimension the space of conformal transformations is infinite dimensional. The space of conformal maps from 2 dimensions into 3 or higher dimensions is even ”bigger” in the sense that it contains non analytic mappings. We shall see for in- stance in section V that any W 2,2 graph in R2 R possess locally × a conformal parametrization. Looking now at transformations (diffeomorphisms) in dimension larger than 2 the constraint of being conformal is suddenly even more rigid and reduces to a finite dimensional group. Precisely we have the following theorem which was first proved for smooth conformal transformation of R3 by Joseph Liouville around 1850. Theorem II.1. Let n > 3 and an open connected set of Rn. Any conformal smooth map (at least W 1,n( )) from into Rn is loc U of the form P (x x ) u(x)= a + α − 0 x x δ | − 0 | Where a Rn V , α R, P O(n), the space of orthogonal ∈ \ ∈ ∈ matrices, and δ is either 0 or 2. ✷ 2More generally for a map u going from a riemann surface Σ into a riemannian manifold (N, h) the cancellation of the intrinsic 1 0 1 0 − ⊗ − Hopf diffenrential given in local holomorphic coordinates z = x1 + ix2 by φ(u)= ∂ u 2 ∂ u 2 2i < ∂ u, ∂ u> dz dz | x1 |h − | x2 | − x1 x2 h ⊗ characterizes the conformality of u. 7 In other words u is either constant or is the translation of the composition of an isometry with a dilation or the composition of an inversion, an isometry and a dilation. Regarding the as- sumption on the regularity, there is a weak formulation for being conformal (see [Iwaniez Martin]) which permits to consider the conformal condition for W 1,1 maps from Rn into Rn. Iwaniez and Martin proved that the rigidity result given by Liouville Theorem still holds for W 1,p( , Rn) maps when p > n U 2 and n is even and they provided counterexamples to Liouville 1,p n n Theorem in W ( , R ) for any p< 2 . Whether the threshold n U 2 holds also for n odd is still open. We shall give a Proof of Theorem II.1 for u being a C4 diffeomorphism. In order to do so we shall first prove the following Lemma. Lemma II.3. Let CRn be open and u be a C4 conformal U diffeomorphism from into u( )CRn then there exists A and U U B in R such that x X,Y Rn ∀ ∈ U ∀ ∈ 1 < du X,du Y >= <X,Y > x · x · (A x x 2 +B)2 | − 0 | where < , > denotes the canonical scalar product in Rn. ✷ · · Proof of Lemma II.3 Denote eλ = ∂u for any i =1...n | ∂xi | λ ∂u n By assumption e− forms an orthonormal basis of R . Thus ∂xi there exist coefficients Xj C2(U) such that ki ∈ 2 n ∂ u j λ ∂u = X e− (II.11) ∂x ∂x ki ∂x k i j=1 j X We have with those notations for any choice of i,j,k 1...n ∈{ } 8 ∂ 2λ ∂ ∂u ∂u (e δij)= < , > ∂xk ∂xk ∂xi ∂xj j λ i λ = Xki e + Xkj e From this identity we deduce i,j,k 1...n ∀ ∈{ } j ∂λ λ ∂λ λ ∂λ λ Xik = δij e + δjk e δik e ∂xk ∂xi − ∂xj (II.12) Combining (II.11) and (II.12) for a pair of indices ∂2u ∂λ ∂u ∂λ ∂u = + k = i ∂xk∂xi ∂xk ∂xi ∂xi ∂xk ∀ 6 n (II.13) ∂2u ∂λ ∂u ∂λ ∂u =2 i 2 ∂x ∂x − ∂x ∂x ∀ ∂xi i i j=1 j j X λ ∂ Multiplying the first line of (II.13) by e− and taking the ∂xj derivative gives λ 2 3 2 λ ∂e− ∂ u λ ∂ u ∂ e− ∂u + e− + ∂xj ∂xk∂xi ∂xk∂xi∂xj ∂xj∂xk ∂xi (II.14) ∂e λ ∂2u ∂e λ ∂2u + − + − =0 ∂xk ∂xi∂xj ∂xi ∂xk∂xj 9 Exchanging the role of i and j assuming moreover that j = k and 6 subtracting to (II.14) the corresponding identity for the triplet (j,i,k) instead of (i,j,k) gives ∂2e λ ∂u ∂2e λ ∂u − − =0 (II.15) ∂xj∂xk ∂xi − ∂xi∂xk ∂xj We are assuming that n > 3 hence, for any pair i = k there 6 exists a j = k and j = i.
Load more

Recommended publications
  • HARMONIC FUNCTIONS for BOUNDED and UNBOUNDED P(X)
    UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 UNIVERSITY OF JYVASKYL¨ A¨ UNIVERSITAT¨ JYVASKYL¨ A¨ DEPARTMENT OF MATHEMATICS INSTITUT FUR¨ MATHEMATIK AND STATISTICS UND STATISTIK REPORT 130 BERICHT 130 EXISTENCE AND UNIQUENESS OF p(x)-HARMONIC FUNCTIONS FOR BOUNDED AND UNBOUNDED p(x) JUKKA KEISALA JYVASKYL¨ A¨ 2011 Editor: Pekka Koskela Department of Mathematics and Statistics P.O. Box 35 (MaD) FI{40014 University of Jyv¨askyl¨a Finland ISBN 978-951-39-4299-1 ISSN 1457-8905 Copyright c 2011, Jukka Keisala and University of Jyv¨askyl¨a University Printing House Jyv¨askyl¨a2011 Contents 0 Introduction 3 0.1 Notation and prerequisities . 6 1 Constant p, 1 < p < 1 8 1.1 The direct method of calculus of variations . 10 1.2 Dirichlet energy integral . 10 2 Infinity harmonic functions, p ≡ 1 13 2.1 Existence of solutions . 14 2.2 Uniqueness of solutions . 18 2.3 Minimizing property and related topics . 22 3 Variable p(x), with 1 < inf p(x) < sup p(x) < +1 24 4 Variable p(x) with p(·) ≡ +1 in a subdomain 30 4.1 Approximate solutions uk ...................... 32 4.2 Passing to the limit . 39 5 One-dimensional case, where p is continuous and sup p(x) = +1 45 5.1 Discussion . 45 5.2 Preliminary results . 47 5.3 Measure of fp(x) = +1g is positive . 48 6 Appendix 51 References 55 0 Introduction In this licentiate thesis we study the Dirichlet boundary value problem ( −∆ u(x) = 0; if x 2 Ω; p(x) (0.1) u(x) = f(x); if x 2 @Ω: Here Ω ⊂ Rn is a bounded domain, p :Ω ! (1; 1] a measurable function, f : @Ω ! R the boundary data, and −∆p(x)u(x) is the p(x)-Laplace operator, which is written as p(x)−2 −∆p(x)u(x) = − div jru(x)j ru(x) for finite p(x).
    [Show full text]
  • Berestycki, Introduction to the Gaussian Free Field and Liouville
    Introduction to the Gaussian Free Field and Liouville Quantum Gravity Nathanaël Berestycki [Draft lecture notes: July 19, 2016] 1 Contents 1 Definition and properties of the GFF6 1.1 Discrete case∗ ...................................6 1.2 Green function..................................8 1.3 GFF as a stochastic process........................... 10 1.4 GFF as a random distribution: Dirichlet energy∗ ................ 13 1.5 Markov property................................. 15 1.6 Conformal invariance............................... 16 1.7 Circle averages.................................. 16 1.8 Thick points.................................... 18 1.9 Exercises...................................... 19 2 Liouville measure 22 2.1 Preliminaries................................... 23 2.2 Convergence and uniform integrability in the L2 phase............ 24 2.3 The GFF viewed from a Liouville typical point................. 25 2.4 General case∗ ................................... 27 2.5 The phase transition in Gaussian multiplicative chaos............. 29 2.6 Conformal covariance............................... 29 2.7 Random surfaces................................. 32 2.8 Exercises...................................... 32 3 The KPZ relation 34 3.1 Scaling exponents; KPZ theorem........................ 34 3.2 Applications of KPZ to exponents∗ ....................... 36 3.3 Proof in the case of expected Minkowski dimension.............. 37 3.4 Sketch of proof of Hausdorff KPZ using circle averages............ 39 3.5 Proof of multifractal spectrum of Liouville
    [Show full text]
  • Properly-Weighted Graph Laplacian for Semi-Supervised Learning
    PROPERLY-WEIGHTED GRAPH LAPLACIAN FOR SEMI-SUPERVISED LEARNING JEFF CALDER Department of Mathematics, University of Minnesota DEJAN SLEPČEV Department of Mathematical Sciences, Carnegie Mellon University Abstract. The performance of traditional graph Laplacian methods for semi-supervised learning degrades substantially as the ratio of labeled to unlabeled data decreases, due to a degeneracy in the graph Laplacian. Several approaches have been proposed recently to address this, however we show that some of them remain ill-posed in the large-data limit. In this paper, we show a way to correctly set the weights in Laplacian regularization so that the estimator remains well posed and stable in the large-sample limit. We prove that our semi-supervised learning algorithm converges, in the infinite sample size limit, to the smooth solution of a continuum variational problem that attains the labeled values continuously. Our method is fast and easy to implement. 1. Introduction For many applications of machine learning, such as medical image classification and speech recognition, labeling data requires human input and is expensive [13], while unlabeled data is relatively cheap. Semi-supervised learning aims to exploit this dichotomy by utilizing the geometric or topological properties of the unlabeled data, in conjunction with the labeled data, to obtain better learning algorithms. A significant portion of the semi-supervised literature is on transductive learning, whereby a function is learned only at the unlabeled points, and not as a parameterized function on an ambient space. In the transductive setting, graph based algorithms, such the graph Laplacian-based learning pioneered by [53], are widely used and have achieved great success [3,26,27,44–48,50, 52].
    [Show full text]
  • A Simple Discretization of the Vector Dirichlet Energy
    Eurographics Symposium on Geometry Processing 2020 Volume 39 (2020), Number 5 Q. Huang and A. Jacobson (Guest Editors) A Simple Discretization of the Vector Dirichlet Energy Oded Stein1, Max Wardetzky2, Alec Jacobson3 and Eitan Grinspun3;1 1Columbia University, USA 2University of Göttingen, Germany 3University of Toronto, Canada Abstract We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix-Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy. 1. Introduction The covariant derivative r generalizes the gradient of scalar func- tions to vector fields defined on surfaces. As the gradient does for scalar functions, the covariant derivative measures the infinitesimal change of a vector field in every direction. As with the gradient’s 1 R 2 scalar Dirichlet energy, Escalar(u) := 2 W kruk dx for a smooth scalar function u and a smooth surface W, the covariant derivative has a corresponding vector Dirichlet energy, Z 1 2 E(u) := krukF dx , (1) transporting the 2 W red vector (enlarged) denoising a vector field by smoothing across the surface where u is a smooth vector-valued function on W, and k·kF is the Frobenius norm.
    [Show full text]
  • The Laplacian in Its Different Guises
    U.U.D.M. Project Report 2019:8 The Laplacian in its different guises Oskar Tegby Examensarbete i matematik, 15 hp Handledare: Martin Strömqvist Examinator: Jörgen Östensson Februari 2019 Department of Mathematics Uppsala University Contents 1 p = 2: The Laplacian 3 1.1 In the Dirichlet problem . 3 1.2 In vector calculus . 6 1.3 As a minimizer . 8 1.4 In the complex plane . 13 1.5 As the mean value property . 16 1.6 Viscosity solutions . 17 1.7 In stochastic processes . 18 2 2 < p < and p = : The p-Laplacian and -Laplacian 20 2.1 Ded∞uction of th∞e p-Laplacian and -Laplaci∞an . 20 2.2 As a minimizer . ∞. 22 2.3 Uniqueness of its solutions . 24 2.4 Existence of its solutions . 25 2.4.1 Method 1: Weak solutions and Sobolev spaces . 25 2.4.2 Method 2: Viscosity solutions and Perron’s method . 27 2.5 In the complex plane . 28 2.5.1 The nonlinear Cauchy-Riemann equations . 28 2.5.2 Quasiconformal maps . 29 2.6 In the asymptotic expansion . 36 2.7 As a Tug of War game with or without drift . 41 2.8 Uniqueness of the solutions . 44 1 Introduction The Laplacian n ∂2u ∆u := u = ∇ · ∇ ∂x2 i=1 i � is a scalar operator which gives the divergence of the gradient of a scalar field. It is prevalent in the famous Dirichlet problem, whose importance cannot be overstated. It entails finding the solution u to the problem ∆u = 0 in Ω �u = g on ∂Ω. The Dirichlet problem is of fundamental importance in mathematics and its applications, and the efforts to solve the problem has led to many revolutionary ideas and important advances in mathematics.
    [Show full text]
  • Arxiv:1703.06851V1
    SYMMETRIES AND CONSERVATION LAWS OF A NONLINEAR SIGMA MODEL WITH GRAVITINO JÜRGEN JOST, ENNO KEßLER, JÜRGEN TOLKSDORF, RUIJUN WU, MIAOMIAO ZHU Abstract. We show that the action functional of the nonlinear sigma model with gravitino considered in [18] is invariant under rescaled conformal transformations, super Weyl transfor- mations and diffeomorphisms. We give a careful geometric explanation how a variation of the metric leads to the corresponding variation of the spinors. In particular cases and despite using only commutative variables, the functional possesses a degenerate super symmetry. The corre- sponding conservation laws lead to a geometric interpretation of the energy-momentum tensor and supercurrent as holomorphic sections of appropriate bundles. 1. Introduction The main motivation for the introduction of the two-dimensional nonlinear supersymmetric sigma model in quantum field theory, or more specifically super gravity and super string theory, are its symmetries, see for instance [3, 9]. Furthermore, as argued in [20], the functional is determined by its symmetries together with suitable bounds on the order of its Euler–Lagrange equations. While super symmetric models are usually formulated using anti-commutative vari- ables, in [18] an analogue of the two-dimensional nonlinear supersymmetric sigma model using only commutative variables was introduced. Here we would like to give a detailed geometric account of the symmetries of the purely commutative model considered in [18]. In physics as well as in geometry, symmetries of a system always have important implications. In particular, differentiable symmetries of a system give rise to conservation laws—this is ba- sically what the famous Noether’s theorem states, see e.g.
    [Show full text]
  • Regularity in the Calculus of Variations
    Regularity in the Calculus of Variations Connor Mooney 1 Contents 1 Introduction 3 2 Scalar Equations 4 2.1 Gradient Bounds Inherited From the Boundary . 7 2.2 De Giorgi-Nash-Moser . 8 2.3 Density Estimates and H¨olderRegularity . 10 2.4 Harnack Inequalities . 11 2.5 Boundary Estimates . 13 2.6 Dirichlet Problem for Minimal Surfaces . 16 2.7 Interior Gradient Estimate for Minimal Surfaces . 18 2.8 Optimal Gradient Bound . 22 2.9 Concluding Remarks on Minimal Surfaces . 24 2.10 Stability in the Calculus of Variations . 25 3 Elliptic Systems 27 3.1 Basics . 27 3.1.1 Existence . 27 3.1.2 Quasiconvexity . 28 3.1.3 Lower Semicontinuity . 29 3.1.4 Rank One Convexity . 31 3.2 Polyconvexity, Quasiconvexity, and Rank One Convexity . 32 3.2.1 Quadratic Functions . 32 3.3 Sverak's Example . 34 3.4 Linear Theory . 35 3.5 Partial Regularity . 37 3.6 Caccioppoli inequality for quasiconvex functionals . 39 3.7 Counterexamples . 40 3.8 Reverse H¨olderInequality . 42 3.9 Special Structure: Radial . 43 3.10 Quasiconformal Mappings . 44 2 1 Introduction In these notes we outline the regularity theory for minimizers in the calculus of variations. The first part of the notes deals with the scalar case, with emphasis on the minimal surface equation. The second deals with vector mappings, which have different regularity properties due to the loss of the maximum principle. These notes are based on parts of courses given by Prof. Ovidiu Savin in Fall 2011 and Spring 2014. 3 2 Scalar Equations Scalar divergence-form equations generally arise by taking minimizers of interesting energy functionals.
    [Show full text]
  • Calculus of Variations Lecture Notes Riccardo Cristoferi May 9 2016
    Calculus of Variations Lecture Notes Riccardo Cristoferi May 9 2016 Contents Introduction 7 Chapter 1. Some classical problems1 1.1. The brachistochrone problem1 1.2. The hanging cable problem3 1.3. Minimal surfaces of revolution4 1.4. The isoperimetric problem5 N Chapter 2. Minimization on R 9 2.1. The one dimensional case9 2.2. The multidimensional case 11 2.3. Local representation close to non degenerate points 12 2.4. Convex functions 14 2.5. On the notion of differentiability 15 Chapter 3. First order necessary conditions for one dimensional scalar functions 17 3.1. The first variation - C2 theory 17 3.2. The first variation - C1 theory 24 3.3. Lagrangians of some special form 29 3.4. Solution of the brachistochrone problem 32 3.5. Problems with free ending points 37 3.6. Isoperimetric problems 38 3.7. Solution of the hanging cable problem 42 3.8. Broken extremals 44 Chapter 4. First order necessary conditions for general functions 47 4.1. The Euler-Lagrange equation 47 4.2. Natural boundary conditions 49 4.3. Inner variations 51 4.4. Isoperimetric problems 53 4.5. Holomic constraints 54 Chapter 5. Second order necessary conditions 57 5.1. Non-negativity of the second variation 57 5.2. The Legendre-Hadamard necessary condition 58 5.3. The Weierstrass necessary condition 59 Chapter 6. Null lagrangians 63 Chapter 7. Lagrange multipliers and eigenvalues of the Laplace operator 67 Chapter 8. Sufficient conditions 75 8.1. Coercivity of the second variation 75 6 CONTENTS 8.2. Jacobi conjugate points 78 8.3.
    [Show full text]
  • Gaussian Free Field, Liouville Quantum Gravity and Gaussian Multiplicative
    Gaussian free field, Liouville quantum gravity and Gaussian multiplicative chaos Nathanaël Berestycki Ellen Powell [Draft lecture notes: March 16, 2021] 1 Contents 1 Definition and properties of the GFF7 1.1 Discrete case...................................7 1.2 Continuous Green function............................ 11 1.3 GFF as a stochastic process........................... 14 1.4 Integration by parts and Dirichlet energy.................... 17 1.5 Reminders about functions spaces........................ 18 1.6 GFF as a random distribution.......................... 22 1.7 Itô’s isometry for the GFF............................ 25 1.8 Cameron–Martin space of the Dirichlet GFF.................. 26 1.9 Markov property................................. 28 1.10 Conformal invariance............................... 29 1.11 Circle averages.................................. 30 1.12 Thick points.................................... 31 1.13 Exercises...................................... 34 2 Liouville measure 36 2.1 Preliminaries................................... 37 2.2 Convergence and uniform integrability in the L2 phase............ 38 2.3 Weak convergence to Liouville measure..................... 40 2.4 The GFF viewed from a Liouville typical point................. 41 2.5 General case.................................... 44 2.6 The phase transition for the Liouville measure................. 46 2.7 Conformal covariance............................... 46 2.8 Random surfaces................................. 48 2.9 Exercises.....................................
    [Show full text]
  • A Discrete Laplace-Beltrami Operator for Simplicial Surfaces
    A discrete Laplace-Beltrami operator for simplicial surfaces Alexander I. Bobenko Boris A. Springborn February 24, 2006 Institut f¨ur Mathematik, Technische Universit¨at Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany [email protected] [email protected] Abstract We define a discrete Laplace-Beltrami operator for simplicial sur- faces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is sim- ilar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangula- tion of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal sur- faces. The definition of the discrete Laplace-Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Keywords: Laplace operator, Delaunay triangulation, Dirichlet energy, arXiv:math/0503219v3 [math.DG] 24 Feb 2006 simplicial surfaces, discrete differential geometry 1 Dirichlet energy of piecewise linear functions Let S be a simplicial surface in 3-dimensional Euclidean space, i.e. a geomet- 3 ric simplicial complex in Ê whose carrier S is a 2-dimensional submanifold, Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.
    [Show full text]
  • Discrete Differential Geometry: an Applied Introduction
    DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane Last updated: February 25, 2021 Contents Chapter 1. Introduction 3 1.1. Disclaimer 6 1.2. Copyright 6 1.3. Acknowledgements 6 Chapter 2. Combinatorial Surfaces 7 2.1. Abstract Simplicial Complex 8 2.2. Anatomy of a Simplicial Complex: Star, Closure, and Link 10 2.3. Simplicial Surfaces 14 2.4. Adjacency Matrices 16 2.5. Halfedge Mesh 17 2.6. Written Exercises 21 2.7. Coding Exercises 26 Chapter 3. A Quick and Dirty Introduction to Differential Geometry 28 3.1. The Geometry of Surfaces 28 3.2. Derivatives and Tangent Vectors 31 3.3. The Geometry of Curves 34 3.4. Curvature of Surfaces 37 3.5. Geometry in Coordinates 41 Chapter 4. A Quick and Dirty Introduction to Exterior Calculus 45 4.1. Exterior Algebra 46 4.2. Examples of Wedge and Star in Rn 52 4.3. Vectors and 1-Forms 54 4.4. Differential Forms and the Wedge Product 58 4.5. Hodge Duality 62 4.6. Differential Operators 67 4.7. Integration and Stokes’ Theorem 73 4.8. Discrete Exterior Calculus 77 Chapter 5. Curvature of Discrete Surfaces 84 5.1. Vector Area 84 5.2. Area Gradient 87 5.3. Volume Gradient 89 5.4. Other Definitions 91 5.5. Gauss-Bonnet 94 5.6. Numerical Tests and Convergence 95 Chapter 6. The Laplacian 100 6.1. Basic Properties 100 6.2. Discretization via FEM 103 6.3. Discretization via DEC 107 1 CONTENTS 2 6.4. Meshes and Matrices 110 6.5. The Poisson Equation 112 6.6.
    [Show full text]
  • On the Existance of Minimizers of the Variable Exponent Dirichlet Energy Integral
    COMMUNICATIONS ON Website: http://AIMsciences.org PURE AND APPLIED ANALYSIS Volume 5, Number 3, September 2006 pp. 415{422 ON THE EXISTANCE OF MINIMIZERS OF THE VARIABLE EXPONENT DIRICHLET ENERGY INTEGRAL Peter A. HastÄ oÄ Department of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland (Communicated by Stefano Bianchini) Abstract. In this note we consider the Dirichlet energy integral in the variable exponent case under minimal assumptions on the exponent. First we show that the Dirichlet energy integral always has a minimizer if the boundary values are in L1. Second, we give an example which shows that if the so-called \jump- condition", known to be su±cient, is violated, then a minimizer need not exist for unbounded boundary values. 1. Introduction. After decades of sporadic research e®orts, the ¯eld of variable exponent function spaces entered into a phase of great activity starting in the late 1990's. The researchers were motivated both by inherent interest in developing techniques that work in non-translation invariant, but otherwise very natural, func- tion spaces, and by interesting applications (see, e.g., [3, 24]). For a survey of the present state of the ¯eld, the reader is referred to the article [8] which also includes a comprehensive bibliography of over a hundred titles on the subject. Two questions can be singled out as having received most attention: integral operators (Hardy{ Littlewood, Riesz, other singular integrals, see, e.g., [6, 7, 22]) and p(x)-Laplacian equations with Dirichlet boundary values. The latter is the subject matter of this paper.
    [Show full text]