DOCSLIB.ORG

Notes on Partial Differential Equations John K. Hunter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Notes on Partial Differential Equations John K. Hunter Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis1 1Revised 6/18/2014. Thanks to Kris Jenssen and Jan Koch for corrections. Supported in part by NSF Grant #DMS-1312342. Abstract. These are notes from a two-quarter class on PDEs that are heavily based on the book Partial Differential Equations by L. C. Evans, together with other sources that are mostly listed in the Bibliography. The notes cover roughly Chapter 2 and Chapters 5–7 in Evans. There is no claim to any originality in the notes, but I hope — for some readers at least — they will provide a useful supplement. Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨older spaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 8 1.7. Convolutions 9 1.8. Derivatives and multi-index notation 10 1.9. Mollifiers 11 1.10. Boundaries of open sets 13 1.11. Change of variables 17 1.12. Divergence theorem 17 1.13. Gronwall’s inequality 18 Chapter 2. Laplace’s equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack’s inequality 31 2.5. Green’s identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 48 3.3. Distributions 51 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 58 3.6. Approximation of Sobolev functions 59 3.7. Sobolev embedding: p<n 59 3.8. Sobolev embedding: p>n 68 3.9. Boundary values of Sobolev functions 71 3.10. Compactness results 73 3.11. Sobolev functions on Ω Rn 75 Appendix ⊂ 77 3.A. Functions 77 3.B. Measures 82 v vi CONTENTS 3.C. Integration 86 Chapter 4. Elliptic PDEs 91 4.1. Weak formulation of the Dirichlet problem 91 4.2. Variational formulation 93 4.3. The space H−1(Ω) 95 1 4.4. The Poincar´einequality for H0 (Ω) 98 4.5. Existence of weak solutions of the Dirichlet problem 99 4.6. General linear, second order elliptic PDEs 101 4.7. The Lax-Milgram theorem and general elliptic PDEs 103 4.8. Compactness of the resolvent 105 4.9. The Fredholm alternative 106 4.10. The spectrum of a self-adjoint elliptic operator 108 4.11. Interior regularity 110 4.12. Boundary regularity 114 4.13. Some further perspectives 116 Appendix 119 4.A. Heat flow 119 4.B. Operators on Hilbert spaces 121 4.C. Difference quotients 124 Chapter 5. The Heat and Schr¨odinger Equations 127 5.1. The initial value problem for the heat equation 127 5.2. Generalized solutions 134 5.3. The Schr¨odinger equation 138 5.4. Semigroups and groups 139 5.5. A semilinear heat equation 152 5.6. The nonlinear Schr¨odinger equation 157 Appendix 166 5.A. The Schwartz space 166 5.B. The Fourier transform 168 5.C. The Sobolev spaces Hs(Rn) 172 5.D. Fractional integrals 173 Chapter 6. Parabolic Equations 177 6.1. The heat equation 177 6.2. Generalsecond-orderparabolicPDEs 178 6.3. Definition of weak solutions 179 6.4. The Galerkin approximation 181 6.5. Existence of weak solutions 183 6.6. A semilinear heat equation 188 6.7. The Navier-Stokes equation 193 Appendix 196 6.A. Vector-valued functions 196 6.B. Hilbert triples 207 Chapter 7. Hyperbolic Equations 211 7.1. The wave equation 211 7.2. Definition of weak solutions 212 7.3. Existence of weak solutions 214 CONTENTS vii 7.4. Continuity of weak solutions 217 7.5. Uniqueness of weak solutions 219 Chapter 8. Friedrich symmetric systems 223 8.1. A BVP for symmetric systems 223 8.2. Boundary conditions 224 8.3. Uniqueness of smooth solutions 225 8.4. Existence of weak solutions 226 8.5. Weak equals strong 227 Bibliography 235 CHAPTER 1 Preliminaries In this chapter, we collect various definitions and theorems for future use. Proofs may be found in the references e.g. [4, 11, 24, 37, 42, 44]. 1.1. Euclidean space Let Rn be n-dimensional Euclidean space. We denote the Euclidean norm of a vector x = (x , x ,...,x ) Rn by 1 2 n ∈ 1/2 x = x2 + x2 + + x2 | | 1 2 ··· n and the inner product of vectorsx = (x1, x2,...,xn), y = (y1,y2,...,yn) by x y = x y + x y + + x y . · 1 1 2 2 ··· n n We denote Lebesgue measure on Rn by dx, and the Lebesgue measure of a set E Rn by E . ⊂If E is a| subset| of Rn, we denote the complement by Ec = Rn E, the closure \ by E, the interior by E◦ and the boundary by ∂E = E E◦. The characteristic function χ : Rn R of E is defined by \ E → 1 if x E, χ (x)= E 0 if x∈ / E. ∈ A set E is bounded if x : x E is bounded in R. A set is connected if it is not the disjoint union of two{| nonempty| ∈ } relatively open subsets. We sometimes refer to a connected open set as a domain. We say that a (nonempty) open set Ω′ is compactly contained in an open set Ω, written Ω′ ⋐ Ω, if Ω′ Ω and Ω′ is compact. If Ω′ ⋐ Ω, then ⊂ dist (Ω′, ∂Ω) = inf x y : x Ω′,y ∂Ω > 0. {| − | ∈ ∈ } 1.2. Spaces of continuous functions Let Ω be an open set in Rn. We denote the space of continuous functions u : Ω R by C(Ω); the space of functions with continuous partial derivatives in Ω of order→ less than or equal to k N by Ck(Ω); and the space of functions with continuous derivatives of all orders∈ by C∞(Ω). Functions in these spaces need not be bounded even if Ω is bounded; for example, (1/x) C∞(0, 1). ∈ If Ω is a bounded open set in Rn, we denote by C(Ω) the space of continuous functions u : Ω R. This is a Banach space with respect to the maximum, or supremum, norm→ u ∞ = sup u(x) . k k x∈Ω | | We denote the support of a continuous function u :Ω Rn by → supp u = x Ω: u(x) =0 . { ∈ 6 } 1 2 1. PRELIMINARIES We denote by Cc(Ω) the space of continuous functions whose support is compactly ∞ contained in Ω, and by Cc (Ω) the space of functions with continuous derivatives of all orders and compact support in Ω. We will sometimes refer to such functions as test functions. n n The completion of Cc(R ) with respect to the uniform norm is the space C0(R ) of continuous functions that approach zero at infinity. (Note that in many places the ∞ notation C0 and C0 is used to denote the spaces of compactly supported functions ∞ that we denote by Cc and Cc .) If Ω is bounded, then we say that a function u : Ω R belongs to Ck(Ω) if it is continuous and its partial derivatives of order less→ than or equal to k are uniformly continuous in Ω, in which case they extend to continuous functions on Ω. The space Ck(Ω) is a Banach space with respect to the norm α u Ck(Ω) = sup ∂ u k k Ω | | |αX|≤k where we use the multi-index notation for partial derivatives explained in Sec- tion 1.8. This norm is finite because the derivatives ∂αu are continuous functions on the compact set Ω. A vector field X :Ω Rm belongs to Ck(Ω) if each of its components belongs → to Ck(Ω). 1.3. H¨older spaces The definition of continuity is not a quantitative one, because it does not say how rapidly the values u(y) of a function approach its value u(x) as y x. The modulus of continuity ω : [0, ] [0, ] of a general continuous function→ u, satisfying ∞ → ∞ u(x) u(y) ω ( x y ) , | − |≤ | − | may decrease arbitrarily slowly. As a result, despite their simple and natural ap- pearance, spaces of continuous functions are often not suitable for the analysis of PDEs, which is almost always based on quantitative estimates. A straightforward and useful way to strengthen the definition of continuity is to require that the modulus of continuity is proportional to a power x y α for some exponent 0 < α 1. Such functions are said to be H¨older continuous,| − or| Lip- schitz continuous if α≤= 1. Roughly speaking, one can think of H¨older continuous functions with exponent α as functions with bounded fractional derivatives of the the order α. Definition 1.1. Suppose that Ω is an open set in Rn and 0 < α 1. A function u : Ω R is uniformly H¨older continuous with exponent α in≤ Ω if the quantity → u(x) u(y) (1.1) [u]α,Ω = sup | − α | x,y ∈ Ω x y x 6= y | − | is finite. A function u :Ω R is locally uniformly H¨older continuous with exponent → ′ ⋐ 0,α α in Ω if [u]α,Ω′ is finite for every Ω Ω. We denote by C (Ω) the space of locally uniformly H¨older continuous functions with exponent α in Ω. If Ω is bounded, we denote by C0,α Ω the space of uniformly H¨older continuous functions with exponent α in Ω. 1.4. Lp SPACES 3 We typically use Greek letters such as α, β both for H¨older exponents and multi-indices; it should be clear from the context which they denote.
Load more

Recommended publications
  • Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
    Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] .
    [Show full text]
  • On Elliptic Partial Differential Equations Annali Della Scuola Normale Superiore Di Pisa, Classe Di Scienze 3E Série, Tome 13, No 2 (1959), P
    ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze L. NIRENBERG On elliptic partial differential equations Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 13, no 2 (1959), p. 115-162 <http://www.numdam.org/item?id=ASNSP_1959_3_13_2_115_0> © Scuola Normale Superiore, Pisa, 1959, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS by L. NIRENBERG (New York) (*) Outline. This series of lectures will tonch on a number of topics in the theory of elliptic differential equations. Iii Lecture I we discuss the fundamental solution for equations with constant coefficients. Lecture .2 is concerned with Calculus inequalities including the well known oues of Sobolev. In le- ctures 3 and 4 we present the Hilbert space approach to the Dirichlet pro- blem for strongly elliptic systems, and describe various inequalities. Lectures 5 and 6 comprise a self contained proof of the well known fact that « weak » solutions of elliptic equations with sufficiently « smooth &#x3E;&#x3E; coefacients are classical solutions. In Lectures 7 and 8 we describe some work of Agmoii, Douglis, Nirenberg [14] concerning estimates near the boundary for solutions of elliptic equations satisfying boundary conditions.
    [Show full text]
  • Notes on Partial Differential Equations John K. Hunter
    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
    [Show full text]
  • MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
    MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and
    [Show full text]
  • Engineering Analysis 2 : Multivariate Functions
    Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Engineering Analysis 2 : Multivariate functions P. Rees, O. Kryvchenkova and P.D. Ledger, [email protected] College of Engineering, Swansea University, UK PDL (CoE) SS 2017 1/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Outline 1 Multivariate functions 2 Partial Differentiation 3 Higher Order Partial Derivatives 4 Total Differentiation 5 Line Integrals 6 Surface Integrals PDL (CoE) SS 2017 2/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction Recall from EG189: analysis of a function of single variable: Differentiation d f (x) d x Expresses the rate of change of a function f with respect to x. Integration Z f (x) d x Reverses the operation of differentiation and can used to work out areas under curves, and as we have just seen to solve ODEs. We’ve seen in vectors we often need to deal with physical fields, that depend on more than one variable. We may be interested in rates of change to each coordinate (e.g. x; y; z), time t, but sometimes also other physical fields as well. PDL (CoE) SS 2017 3/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 1: the area A of a rectangular plate of width x and breadth y can be calculated A = xy The variables x and y are independent of each other. In this case, the dependent variable A is a function of the two independent variables x and y as A = f (x; y) or A ≡ A(x; y) PDL (CoE) SS 2017 4/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 2: the volume of a rectangular plate is given by V = xyz where the thickness of the plate is in z-direction.
    [Show full text]
  • Elliptic Pdes
    viii CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω, (4.2) u =0 on ∂Ω. First, suppose that the boundary of Ω is smooth and u,f : Ω → R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get ∞ (4.3) Du · Dφ dx = fφ dx for all φ ∈ Cc (Ω). ZΩ ZΩ The boundary terms vanish because φ = 0 on the boundary.
    [Show full text]
  • Elliptic Pdes
    CHAPTER 4 Elliptic PDEs One of the main advantages of extending the class of solutions of a PDE from classical solutions with continuous derivatives to weak solutions with weak deriva- tives is that it is easier to prove the existence of weak solutions. Having estab- lished the existence of weak solutions, one may then study their properties, such as uniqueness and regularity, and perhaps prove under appropriate assumptions that the weak solutions are, in fact, classical solutions. There is often considerable freedom in how one defines a weak solution of a PDE; for example, the function space to which a solution is required to belong is not given a priori by the PDE itself. Typically, we look for a weak formulation that reduces to the classical formulation under appropriate smoothness assumptions and which is amenable to a mathematical analysis; the notion of solution and the spaces to which solutions belong are dictated by the available estimates and analysis. 4.1. Weak formulation of the Dirichlet problem Let us consider the Dirichlet problem for the Laplacian with homogeneous boundary conditions on a bounded domain Ω in Rn, (4.1) −∆u = f in Ω; (4.2) u = 0 on @Ω: First, suppose that the boundary of Ω is smooth and u; f : Ω ! R are smooth functions. Multiplying (4.1) by a test function φ, integrating the result over Ω, and using the divergence theorem, we get Z Z 1 (4.3) Du · Dφ dx = fφ dx for all φ 2 Cc (Ω): Ω Ω The boundary terms vanish because φ = 0 on the boundary.
    [Show full text]
  • Advanced Partial Differential Equations Prof. Dr. Thomas
    Advanced Partial Differential Equations Prof. Dr. Thomas Sørensen summer term 2015 Marcel Schaub July 2, 2015 1 Contents 0 Recall PDE 1 & Motivation 3 0.1 Recall PDE 1 . .3 1 Weak derivatives and Sobolev spaces 7 1.1 Sobolev spaces . .8 1.2 Approximation by smooth functions . 11 1.3 Extension of Sobolev functions . 13 1.4 Traces . 15 1.5 Sobolev inequalities . 17 2 Linear 2nd order elliptic PDE 25 2.1 Linear 2nd order elliptic partial differential operators . 25 2.2 Weak solutions . 26 2.3 Existence via Lax-Milgram . 28 2.4 Inhomogeneous bounday value problems . 35 2.5 The space H−1(U) ................................ 36 2.6 Regularity of weak solutions . 39 A Tutorials 58 A.1 Tutorial 1: Review of Integration . 58 A.2 Tutorial 2 . 59 A.3 Tutorial 3: Norms . 61 A.4 Tutorial 4 . 62 A.5 Tutorial 6 (Sheet 7) . 65 A.6 Tutorial 7 . 65 A.7 Tutorial 9 . 67 A.8 Tutorium 11 . 67 B Solutions of the problem sheets 70 B.1 Solution to Sheet 1 . 70 B.2 Solution to Sheet 2 . 71 B.3 Solution to Problem Sheet 3 . 73 B.4 Solution to Problem Sheet 4 . 76 B.5 Solution to Exercise Sheet 5 . 77 B.6 Solution to Exercise Sheet 7 . 81 B.7 Solution to problem sheet 8 . 84 B.8 Solution to Exercise Sheet 9 . 87 2 0 Recall PDE 1 & Motivation 0.1 Recall PDE 1 We mainly studied linear 2nd order equations – specifically, elliptic, parabolic and hyper- bolic equations. Concretely: • The Laplace equation ∆u = 0 (elliptic) • The Poisson equation −∆u = f (elliptic) • The Heat equation ut − ∆u = 0, ut − ∆u = f (parabolic) • The Wave equation utt − ∆u = 0, utt − ∆u = f (hyperbolic) We studied (“main motivation; goal”) well-posedness (à la Hadamard) 1.
    [Show full text]
  • Arxiv:0906.4883V2 [Math.CA] 4 Jan 2010 Ro O Olna Yeblcsses[4.W Hwblwta El’ T Helly’S That Theorem
    THE KOLMOGOROV–RIESZ COMPACTNESS THEOREM HARALD HANCHE-OLSEN AND HELGE HOLDEN Abstract. We show that the Arzelà–Ascoli theorem and Kolmogorov com- pactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly’s theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem. 1. Introduction Compactness results in the spaces Lp(Rd) (1 p < ) are often vital in exis- tence proofs for nonlinear partial differential equations.≤ ∞ A necessary and sufficient condition for a subset of Lp(Rd) to be compact is given in what is often called the Kolmogorov compactness theorem, or Fréchet–Kolmogorov compactness theorem. Proofs of this theorem are frequently based on the Arzelà–Ascoli theorem. We here show how one can deduce both the Kolmogorov compactness theorem and the Arzelà–Ascoli theorem from one common lemma on compactness in metric spaces, which again is based on the fact that a metric space is compact if and only if it is complete and totally bounded. Furthermore, we trace out the historical roots of Kolmogorov’s compactness theorem, which originated in Kolmogorov’s classical paper [18] from 1931. However, there were several other approaches to the issue of describing compact subsets of Lp(Rd) prior to and after Kolmogorov, and several of these are described in Section 4. Furthermore, extensions to other spaces, say Lp(Rd) (0 p < 1), Orlicz spaces, or compact groups, are described. Helly’s theorem is often≤ used as a replacement for Kolmogorov’s compactness theorem, in particular in the context of nonlinear hyperbolic conservation laws, in spite of being more specialized (e.g., in the sense that its classical version requires one spatial dimension).
    [Show full text]
  • Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing
    Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing By the Faculty of Mathematics and Computer Science of the Technische Universität Bergakademie Freiberg approved Thesis to attain the academic degree of doctor rerum naturalium (Dr. rer. nat.) submitted by Dipl. Math. Martin Reinhardt born on the 1st December, 1987 in Suhl Assessors: Prof. Dr. Swanhild Bernstein Prof. Dr. Konrad Froitzheim Date of the award: Freiberg, 6th March, 2019 Versicherung Hiermit versichere ich, dass ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Hilfe eines Promotionsberaters habe ich nicht in Anspruch genommen. Weitere Personen haben von mir keine geldwerten Leistungen für Arbeiten erhalten, die nicht als solche kenntlich gemacht worden sind. Die Arbeit wurde bisher weder im Inland noch im Ausland in gleicher oder ähnlicher Form einer anderen Prüfungsbehörde vorgelegt. 6. Dezember 2018 Dipl. Math. Martin Reinhardt Declaration I hereby declare that I completed this work without any improper help from a third party and without using any aids other than those cited. All ideas derived directly or indirectly from other sources are identified as such. I did not seek the help of a professional doctorate-consultant. Only those persons identified as having done so received any financial payment from me for any work done for me. This thesis has not previously been published in the same or a similar form in Germany or abroad. 6th March, 2019 Dipl. Math. Martin Reinhardt 5 Contents Preface 7 Introduction 9 1.
    [Show full text]
  • Elliptic Boundary Problems
    CHAPTER 8 Elliptic boundary problems Summary Elliptic boundary problems are discussed, especially for operators of Dirac type. We start with a discussion of manifolds with boundary, including functions spaces and distributions. This leads to the `jumps formula' for the relationship of the action of a differential operator to the operation of cutting off at the boundary; this is really Green's formula. The idea behind Calder`on'sapproach to boundary problems is introduced in the restricted context of a dividing hypersurface in a manifold without boundary. This includes the fundamental result on the boundary behaviour of a pseudodifferential operator with a rational symbol. These ideas are then extended to the case of an operator of Dirac type on a compact manifold with boundary with the use of left and right parametrices to define the Calder`on projector. General boundary problems defined by pseudodifferential projections are discussed by reference to the ‘Calder`onrealization' of the operator. Local boundary conditions, and the corresponding ellipticity conditions, are then discussed and the special case of Hodge theory on a compact manifold with boundary is analysed in detail for absolute and relative boundary conditions. Introduction Elliptic boundary problems arise from the fact that elliptic differential operators on compact manifolds with boundary have infinite dimensional null spaces. The main task we carry out below is the parameterization of this null space, in terms of boundary values, of an elliptic differential operator on a manifold with boundary. For simplicity of presentation the discussion of elliptic boundary problems here will be largely confined to the case of first order systems of differential operators of Dirac type.
    [Show full text]
  • The Lp Resolvents of Elliptic Operators with Uniformly Continuous Coefficients
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Differential Equations 188 (2003) 555–568 The Lp resolvents of elliptic operators with uniformly continuous coefficients Yoichi Miyazaki*,1 School of Dentistry, Nihon University, 8-13, Kanda-Surugadai 1-chome, Chiyoda-ku, Tokyo 101-8310, Japan Received September 19, 2001; revised June 13, 2002 Abstract We consider the strongly elliptic operator A of order 2m in the divergence form with bounded measurable coefficients and assume that the coefficients of top order are uniformly continuous. For 1opoN; A is a bounded linear operator from the Lp Sobolev space Hm;p into HÀm;p: We will prove that ðA À lÞÀ1 exists in HÀm;p for some l and estimate its operator norm. r 2002 Elsevier Science (USA). All rights reserved. Keywords: Lp theory; Elliptic operator; Non-smooth coefficient; Divergence form; Resolvent; Uniformly continuous 1. Introduction Let us consider the elliptic operator defined in the n-dimensional Euclidean space Rn in the divergence form X a b AuðxÞ¼ D ðaabðxÞD uðxÞÞ; ð1:1Þ jaj;jbjpm n where x ¼ðx1; y; xnÞ is a generic point in R and we use the notations pffiffiffiffiffiffiffi a a1 ? an D ¼ D1 Dn ; Dj ¼ À1@=@xj *Corresponding author. E-mail address: [email protected]. 1 Supported partly by Sato Fund, Nihon University School of Dentistry. 0022-0396/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 0396(02)00109-2 556 Y.
    [Show full text]