280 11 Hopf Variational Solutions 8.3 are Weak Solutions of the Cauchy Problem 6.4.. 302 5 Solving the Homogeneous Dirichlet Problem 4.1 by the Riesz Representation Theorem.. 302 6 Solvi
Trang 3Series Editors
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Steven G Krantz, Washington University, St Louis
Advisory Board
Anthony W Knapp, State University of New York at Stony Brook, Emeritus
Trang 5proprietary rights
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Mathematics Subject Classification (2000): 31B05, 31B20, 35A10, 35B45, 35B65, 35D10, 35J05, 35K05, 35L05, 35L60, 35L65, 45A05, 45B05, 45C05, 49J40
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Trang 6Preface to the Second Edition xvii
Preface to the First Edition xix
0 Preliminaries 1
1 Green’s Theorem 1
1.1 Differential Operators and Adjoints 2
2 The Continuity Equation 3
3 The Heat Equation and the Laplace Equation 5
3.1 Variable Coefficients 5
4 A Model for the Vibrating String 6
5 Small Vibrations of a Membrane 8
6 Transmission of Sound Waves 11
7 The Navier–Stokes System 13
8 The Euler Equations 13
9 Isentropic Potential Flows 14
9.1 Steady Potential Isentropic Flows 15
10 Partial Differential Equations 16
1 Quasi-Linear Equations and the Cauchy–Kowalewski Theorem 17
1 Quasi-Linear Second-Order Equations in Two Variables 17
2 Characteristics and Singularities 19
2.1 Coefficients Independent of u x and u y 20
3 Quasi-Linear Second-Order Equations 21
3.1 Constant Coefficients 23
3.2 Variable Coefficients 23
4 Quasi-Linear Equations of Order m ≥ 1 24
4.1 Characteristic Surfaces 25
5 Analytic Data and the Cauchy–Kowalewski Theorem 26
5.1 Reduction to Normal Form ([19]) 26
Trang 7vi Contents
6 Proof of the Cauchy–Kowalewski Theorem 27
6.1 Estimating the Derivatives of u at the Origin 28
7 Auxiliary Inequalities 29
8 Auxiliary Estimations at the Origin 31
9 Proof of the Cauchy–Kowalewski Theorem (Concluded) 32
9.1 Proof of Lemma 6.1 33
Problems and Complements 33
1c Quasi-Linear Second-Order Equations in Two Variables 33
5c Analytic Data and the Cauchy–Kowalewski Theorem 34
6c Proof of the Cauchy–Kowalewski Theorem 34
8c The Generalized Leibniz Rule 34
9c Proof of the Cauchy–Kowalewski Theorem (Concluded) 35
2 The Laplace Equation 37
1 Preliminaries 37
1.1 The Dirichlet and Neumann Problems 38
1.2 The Cauchy Problem 39
1.3 Well-Posedness and a Counterexample of Hadamard 39
1.4 Radial Solutions 40
2 The Green and Stokes Identities 41
2.1 The Stokes Identities 41
3 Green’s Function and the Dirichlet Problem for a Ball 43
3.1 Green’s Function for a Ball 45
4 Sub-Harmonic Functions and the Mean Value Property 47
4.1 The Maximum Principle 50
4.2 Structure of Sub-Harmonic Functions 50
5 Estimating Harmonic Functions and Their Derivatives 52
5.1 The Harnack Inequality and the Liouville Theorem 52
5.2 Analyticity of Harmonic Functions 53
6 The Dirichlet Problem 55
7 About the Exterior Sphere Condition 58
7.1 The Case N = 2 and ∂E Piecewise Smooth 59
7.2 A Counterexample of Lebesgue for N = 3 ([101]) 59
8 The Poisson Integral for the Half-Space 60
9 Schauder Estimates of Newtonian Potentials 62
10 Potential Estimates in L p (E) 65
11 Local Solutions 68
11.1 Local Weak Solutions 69
12 Inhomogeneous Problems 70
12.1 On the Notion of Green’s Function 70
12.2 Inhomogeneous Problems 71
12.3 The Case f ∈ C ∞ o (E) 72
12.4 The Case f ∈ C η( ¯E) 72
Trang 8Problems and Complements 73
1c Preliminaries 73
1.1c Newtonian Potentials on Ellipsoids 73
1.2c Invariance Properties 74
2c The Green and Stokes Identities 74
3c Green’s Function and the Dirichlet Problem for the Ball 74
3.1c Separation of Variables 75
4c Sub-Harmonic Functions and the Mean Value Property 76
4.1c Reflection and Harmonic Extension 77
4.2c The Weak Maximum Principle 77
4.3c Sub-Harmonic Functions 78
5c Estimating Harmonic Functions 79
5.1c Harnack-Type Estimates 80
5.2c Ill-Posed Problems: An Example of Hadamard 80
5.3c Removable Singularities 81
7c About the Exterior Sphere Condition 82
8c Problems in Unbounded Domains 83
8.1c The Dirichlet Problem Exterior to a Ball 83
9c Schauder Estimates up to the Boundary ([135, 136]) 84
10c Potential Estimates in L p (E) 84
10.1c Integrability of Riesz Potentials 85
10.2c Second Derivatives of Potentials 85
3 Boundary Value Problems by Double-Layer Potentials 87
1 The Double-Layer Potential 87
2 On the Integral Defining the Double-Layer Potential 89
3 The Jump Condition of W (∂E, x o ; v) Across ∂E 91
4 More on the Jump Condition Across ∂E 93
5 The Dirichlet Problem by Integral Equations ([111]) 94
6 The Neumann Problem by Integral Equations ([111]) 95
7 The Green Function for the Neumann Problem 97
7.1 Finding G(·; ·) 98
8 Eigenvalue Problems for the Laplacian 99
8.1 Compact Kernels Generated by Green’s Function 100
9 Compactness of A F in L p (E) for 1 ≤ p ≤ ∞ 100
10 Compactness of A Φ in L p (E) for 1 ≤ p < ∞ 102
11 Compactness of A Φ in L ∞ (E) 102
Problems and Complements 104
2c On the Integral Defining the Double-Layer Potential 104
5c The Dirichlet Problem by Integral Equations 105
6c The Neumann Problem by Integral Equations 106
Trang 9viii Contents
7c Green’s Function for the Neumann Problem 106
7.1c ConstructingG(·; ·) for a Ball in R2 andR3 106
8c Eigenvalue Problems 107
4 Integral Equations and Eigenvalue Problems 109
1 Kernels in L2(E) 109
1.1 Examples of Kernels in L2(E) 110
2 Integral Equations in L2(E) 111
2.1 Existence of Solutions for Small|λ| 111
3 Separable Kernels 112
3.1 Solving the Homogeneous Equations 113
3.2 Solving the Inhomogeneous Equation 113
4 Small Perturbations of Separable Kernels 114
4.1 Existence and Uniqueness of Solutions 115
5 Almost Separable Kernels and Compactness 116
5.1 Solving Integral Equations for Almost Separable Kernels 117
5.2 Potential Kernels Are Almost Separable 117
6 Applications to the Neumann Problem 118
7 The Eigenvalue Problem 119
8 Finding a First Eigenvalue and Its Eigenfunctions 121
9 The Sequence of Eigenvalues 122
9.1 An Alternative Construction Procedure of the Sequence of Eigenvalues 123
10 Questions of Completeness and the Hilbert–Schmidt Theorem 124 10.1 The Case of K(x; ·) ∈ L2(E) Uniformly in x 125
11 The Eigenvalue Problem for the Laplacean 126
11.1 An Expansion of Green’s Function 127
Problems and Complements 128
2c Integral Equations 128
2.1c Integral Equations of the First Kind 128
2.2c Abel Equations ([2, 3]) 128
2.3c Solving Abel Integral Equations 129
2.4c The Cycloid ([3]) 130
2.5c Volterra Integral Equations ([158, 159]) 130
3c Separable Kernels 131
3.1c Hammerstein Integral Equations ([64]) 131
6c Applications to the Neumann Problem 132
9c The Sequence of Eigenvalues 132
10c Questions of Completeness 132
10.1c Periodic Functions inRN 133
10.2c The Poisson Equation with Periodic Boundary Conditions 134
11c The Eigenvalue Problem for the Laplacian 134
Trang 105 The Heat Equation 135
1 Preliminaries 135
1.1 The Dirichlet Problem 136
1.2 The Neumann Problem 136
1.3 The Characteristic Cauchy Problem 136
2 The Cauchy Problem by Similarity Solutions 136
2.1 The Backward Cauchy Problem 140
3 The Maximum Principle and Uniqueness (Bounded Domains) 140 3.1 A Priori Estimates 141
3.2 Ill-Posed Problems 141
3.3 Uniqueness (Bounded Domains) 142
4 The Maximum Principle inRN 142
4.1 A Priori Estimates 144
4.2 About the Growth Conditions (4.3) and (4.4) 145
5 Uniqueness of Solutions to the Cauchy Problem 145
5.1 A Counterexample of Tychonov ([155]) 145
6 Initial Data in L1loc(RN) 147
6.1 Initial Data in the Sense of L1loc(RN) 149
7 Remarks on the Cauchy Problem 149
7.1 About Regularity 149
7.2 Instability of the Backward Problem 150
8 Estimates Near t = 0 151
9 The Inhomogeneous Cauchy Problem 152
10 Problems in Bounded Domains 154
10.1 The Strong Solution 155
10.2 The Weak Solution and Energy Inequalities 156
11 Energy and Logarithmic Convexity 157
11.1 Uniqueness for Some Ill-Posed Problems 158
12 Local Solutions 158
12.1 Variable Cylinders 162
12.2 The Case|α| = 0 162
13 The Harnack Inequality 163
13.1 Compactly Supported Sub-Solutions 164
13.2 Proof of Theorem 13.1 165
14 Positive Solutions in S T 167
14.1 Non-Negative Solutions 169
Problems and Complements 171
2c Similarity Methods 171
2.1c The Heat Kernel Has Unit Mass 171
2.2c The Porous Media Equation 172
2.3c The p-Laplacean Equation 172
2.4c The Error Function 173
2.5c The Appell Transformation ([7]) 173
2.6c The Heat Kernel by Fourier Transform 173
Trang 11x Contents
2.7c Rapidly Decreasing Functions 174
2.8c The Fourier Transform of the Heat Kernel 174
2.9c The Inversion Formula 175
3c The Maximum Principle in Bounded Domains 176
3.1c The Blow-Up Phenomenon for Super-Linear Equations 177 3.2c The Maximum Principle for General Parabolic Equations 178
4c The Maximum Principle inRN 178
4.1c A Counterexample of the Tychonov Type 180
7c Remarks on the Cauchy Problem 180
12c On the Local Behavior of Solutions 180
6 The Wave Equation 183
1 The One-Dimensional Wave Equation 183
1.1 A Property of Solutions 184
2 The Cauchy Problem 185
3 Inhomogeneous Problems 186
4 A Boundary Value Problem (Vibrating String) 188
4.1 Separation of Variables 189
4.2 Odd Reflection 190
4.3 Energy and Uniqueness 190
4.4 Inhomogeneous Problems 191
5 The Initial Value Problem in N Dimensions 191
5.1 Spherical Means 192
5.2 The Darboux Formula 192
5.3 An Equivalent Formulation of the Cauchy Problem 193
6 The Cauchy Problem inR3 193
7 The Cauchy Problem inR2 196
8 The Inhomogeneous Cauchy Problem 198
9 The Cauchy Problem for Inhomogeneous Surfaces 199
9.1 Reduction to Homogeneous Data on t = Φ 200
9.2 The Problem with Homogeneous Data 200
10 Solutions in Half-Space The Reflection Technique 201
10.1 An Auxiliary Problem 202
10.2 Homogeneous Data on the Hyperplane x3= 0 202
11 A Boundary Value Problem 203
12 Hyperbolic Equations in Two Variables 204
13 The Characteristic Goursat Problem 205
13.1 Proof of Theorem 13.1: Existence 205
13.2 Proof of Theorem 13.1: Uniqueness 207
13.3 Goursat Problems in Rectangles 207
14 The Non-Characteristic Cauchy Problem and the Riemann Function 208
15 Symmetry of the Riemann Function 210
Trang 12Problems and Complements 211
2c The d’Alembert Formula 211
3c Inhomogeneous Problems 211
3.1c The Duhamel Principle ([38]) 211
4c Solutions for the Vibrating String 212
6c Cauchy Problems inR3 214
6.1c Asymptotic Behavior 214
6.2c Radial Solutions 214
6.3c Solving the Cauchy Problem by Fourier Transform 216
7c Cauchy Problems inR2 and the Method of Descent 217
7.1c The Cauchy Problem for N = 4, 5 218
8c Inhomogeneous Cauchy Problems 218
8.1c The Wave Equation for the N and (N + 1)-Laplacian 218
8.2c Miscellaneous Problems 219
10c The Reflection Technique 221
11c Problems in Bounded Domains 221
11.1c Uniqueness 221
11.2c Separation of Variables 222
12c Hyperbolic Equations in Two Variables 222
12.1c The General Telegraph Equation 222
14c Goursat Problems 223
14.1c The Riemann Function and the Fundamental Solution of the Heat Equation 223
7 Quasi-Linear Equations of First-Order 225
1 Quasi-Linear Equations 225
2 The Cauchy Problem 226
2.1 The Case of Two Independent Variables 226
2.2 The Case of N Independent Variables 227
3 Solving the Cauchy Problem 227
3.1 Constant Coefficients 228
3.2 Solutions in Implicit Form 229
4 Equations in Divergence Form and Weak Solutions 230
4.1 Surfaces of Discontinuity 231
4.2 The Shock Line 231
5 The Initial Value Problem 232
5.1 Conservation Laws 233
6 Conservation Laws in One Space Dimension 234
6.1 Weak Solutions and Shocks 235
6.2 Lack of Uniqueness 236
7 Hopf Solution of The Burgers Equation 236
8 Weak Solutions to (6.4) When a( ·) is Strictly Increasing 238
8.1 Lax Variational Solution 239
9 Constructing Variational Solutions I 240
9.1 Proof of Lemma 9.1 241
Trang 13xii Contents
10 Constructing Variational Solutions II 242
11 The Theorems of Existence and Stability 244
11.1 Existence of Variational Solutions 244
11.2 Stability of Variational Solutions 245
12 Proof of Theorem 11.1 246
12.1 The Representation Formula (11.4) 246
12.2 Initial Datum in the Sense of L1 loc(R) 247
12.3 Weak Forms of the PDE 248
13 The Entropy Condition 248
13.1 Entropy Solutions 249
13.2 Variational Solutions of (6.4) are Entropy Solutions 249
13.3 Remarks on the Shock and the Entropy Conditions 251
14 The Kruzhkov Uniqueness Theorem 253
14.1 Proof of the Uniqueness Theorem I 253
14.2 Proof of the Uniqueness Theorem II 254
14.3 Stability in L1(RN) 256
15 The Maximum Principle for Entropy Solutions 256
Problems and Complements 257
3c Solving the Cauchy Problem 257
6c Explicit Solutions to the Burgers Equation 259
6.2c Invariance of Burgers Equations by Some Transformation of Variables 259
6.3c The Generalized Riemann Problem 260
13c The Entropy Condition 261
14c The Kruzhkov Uniqueness Theorem 262
8 Non-Linear Equations of First-Order 265
1 Integral Surfaces and Monge’s Cones 265
1.1 Constructing Monge’s Cones 266
1.2 The Symmetric Equation of Monge’s Cones 266
2 Characteristic Curves and Characteristic Strips 267
2.1 Characteristic Strips 268
3 The Cauchy Problem 269
3.1 Identifying the Initial Data p(0, s) 269
3.2 Constructing the Characteristic Strips 270
4 Solving the Cauchy Problem 270
4.1 Verifying (4.3) 271
4.2 A Quasi-Linear Example inR2 272
5 The Cauchy Problem for the Equation of Geometrical Optics 273
5.1 Wave Fronts, Light Rays, Local Solutions, and Caustics 274
6 The Initial Value Problem for Hamilton–Jacobi Equations 274
7 The Cauchy Problem in Terms of the Lagrangian 276
Trang 148 The Hopf Variational Solution 277
8.1 The First Hopf Variational Formula 278
8.2 The Second Hopf Variational Formula 278
9 Semigroup Property of Hopf Variational Solutions 279
10 Regularity of Hopf Variational Solutions 280
11 Hopf Variational Solutions (8.3) are Weak Solutions of the Cauchy Problem (6.4) 281
12 Some Examples 283
12.1 Example I 283
12.2 Example II 284
12.3 Example III 284
13 Uniqueness 285
14 More on Uniqueness and Stability 287
14.1 Stability in L p(RN ) for All p ≥ 1 287
14.2 Comparison Principle 288
15 Semi-Concave Solutions of the Cauchy Problem 288
15.1 Uniqueness of Semi-Concave Solutions 288
16 A Weak Notion of Semi-Concavity 289
17 Semi-Concavity of Hopf Variational Solutions 290
17.1 Weak Semi-Concavity of Hopf Variational Solutions Induced by the Initial Datum u o 290
17.2 Strictly Convex Hamiltonian 291
18 Uniqueness of Weakly Semi-Concave Variational Hopf Solutions 293
9 Linear Elliptic Equations with Measurable Coefficients 297
1 Weak Formulations and Weak Derivatives 297
1.1 Weak Derivatives 298
2 Embeddings of W 1,p (E) 299
2.1 Compact Embeddings of W 1,p (E) 300
3 Multiplicative Embeddings of W 1,p o (E) and ˜ W 1,p (E) 300
3.1 Some Consequences of the Multiplicative Embedding Inequalities 301
4 The Homogeneous Dirichlet Problem 302
5 Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem 302
6 Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods 303
6.1 The Case N = 2 304
6.2 Gˆateaux Derivative and The Euler Equation of J ( ·) 305
7 Solving the Homogeneous Dirichlet Problem (4.1) by Galerkin Approximations 305
7.1 On the Selection of an Orthonormal System in W 1,2 o (E) 306
Trang 15xiv Contents
7.2 Conditions on f and f for the Solvability of the
Dirichlet Problem (4.1) 307
8 Traces on ∂E of Functions in W 1,p (E) 307
8.1 The Segment Property 307
8.2 Defining Traces 308
8.3 Characterizing the Traces on ∂E of Functions in W 1,p (E) 309
9 The Inhomogeneous Dirichlet Problem 309
10 The Neumann Problem 310
10.1 A Variant of (10.1) 311
11 The Eigenvalue Problem 312
12 Constructing the Eigenvalues of (11.1) 313
13 The Sequence of Eigenvalues and Eigenfunctions 315
14 A Priori L ∞ (E) Estimates for Solutions of the Dirichlet Problem (9.1) 317
15 Proof of Propositions 14.1–14.2 318
15.1 An Auxiliary Lemma on Fast Geometric Convergence 319
15.2 Proof of Proposition 14.1 for N > 2 319
15.3 Proof of Proposition 14.1 for N = 2 320
16 A Priori L ∞ (E) Estimates for Solutions of the Neumann Problem (10.1) 320
17 Proof of Propositions 16.1–16.2 322
17.1 Proof of Proposition 16.1 for N > 2 324
17.2 Proof of Proposition 16.1 for N = 2 325
18 Miscellaneous Remarks on Further Regularity 325
Problems and Complements 326
1c Weak Formulations and Weak Derivatives 326
1.1c The Chain Rule in W 1,p (E) 326
2c Embeddings of W 1,p (E) 327
2.1c Proof of (2.4) 327
2.2c Compact Embeddings of W 1,p (E) 328
3c Multiplicative Embeddings of W o 1,p (E) and ˜ W 1,p (E) 329
3.1c Proof of Theorem 3.1 for 1≤ p < N 329
3.2c Proof of Theorem 3.1 for p ≥ N > 1 331
3.3c Proof of Theorem 3.2 for 1≤ p < N and E Convex 332
5c Solving the Homogeneous Dirichlet Problem (4.1) by the Riesz Representation Theorem 333
6c Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods 334
6.1c More General Variational Problems 334
6.8c Gˆateaux Derivatives, Euler Equations, and Quasi-Linear Elliptic Equations 336
Trang 168c Traces on ∂E of Functions in W 1,p (E) 337
8.1c Extending Functions in W 1,p (E) 337
8.2c The Trace Inequality 338
8.3c Characterizing the Traces on ∂E of Functions in W 1,p (E) 339
9c The Inhomogeneous Dirichlet Problem 341
9.1c The Lebesgue Spike 341
9.2c Variational Integrals and Quasi-Linear Equations 341
10c The Neumann Problem 342
11c The Eigenvalue Problem 343
12c Constructing the Eigenvalues 343
13c The Sequence of Eigenvalues and Eigenfunctions 343
14c A Priori L ∞ (E) Estimates for Solutions of the Dirichlet Problem (9.1) 343
15c A Priori L ∞ (E) Estimates for Solutions of the Neumann Problem (10.1) 344
15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c) 344
10 DeGiorgi Classes 347
1 Quasi-Linear Equations and DeGiorgi Classes 347
1.1 DeGiorgi Classes 349
2 Local Boundedness of Functions in the DeGiorgi Classes 350
2.1 Proof of Theorem 2.1 for 1 < p < N 351
2.2 Proof of Theorem 2.1 for p = N 352
3 H¨older Continuity of Functions in the DG Classes 353
3.1 On the Proof of Theorem 3.1 354
4 Estimating the Values of u by the Measure of the Set where u is Either Near μ+ or Near μ − 354
5 Reducing the Measure of the Set where u is Either Near μ+ or Near μ − 355
5.1 The Discrete Isoperimetric Inequality 356
5.2 Proof of Proposition 5.1 357
6 Proof of Theorem 3.1 358
7 Boundary DeGiorgi Classes: Dirichlet Data 359
7.1 Continuity up to ∂E of Functions in the Boundary DG Classes (Dirichlet Data) 360
8 Boundary DeGiorgi Classes: Neumann Data 361
8.1 Continuity up to ∂E of Functions in the Boundary DG Classes (Neumann Data) 363
9 The Harnack Inequality 364
9.1 Proof of Theorem 9.1 (Preliminaries) 364
9.2 Proof of Theorem 9.1 Expansion of Positivity 365
9.3 Proof of Theorem 9.1 365
10 Harnack Inequality and H¨older Continuity 367
Trang 18This is a revised and extended version of my 1995 elementary introduction
to partial differential equations The material is essentially the same exceptfor three new chapters The first (Chapter 8) is about non-linear equations
of first order and in particular Hamilton–Jacobi equations It builds on thecontinuing idea that PDEs, although a branch of mathematical analysis, areclosely related to models of physical phenomena Such underlying physics
in turn provides ideas of solvability The Hopf variational approach to theCauchy problem for Hamilton–Jacobi equations is one of the clearest andmost incisive examples of such an interplay The method is a perfect blend
of classical mechanics, through the role and properties of the Lagrangian andHamiltonian, and calculus of variations A delicate issue is that of identifying
“uniqueness classes.” An effort has been made to extract the geometricalconditions on the graph of solutions, such as quasi-concavity, for uniqueness
to hold
Chapter 9 is an introduction to weak formulations, Sobolev spaces, anddirect variational methods for linear and quasi-linear elliptic equations Whileterse, the material on Sobolev spaces is reasonably complete, at least for aPDE user It includes all the basic embedding theorems, including their proofs,and the theory of traces Weak formulations of the Dirichlet and Neumannproblems build on this material Related variational and Galerkin methods,
as well as eigenvalue problems, are presented within their weak framework.The Neumann problem is not as frequently treated in the literature as theDirichlet problem; an effort has been made to present the underlying theory
as completely as possible Some attention has been paid to the local behavior
of these weak solutions, both for the Dirichlet and Neumann problems Whileefficient in terms of existence theory, weak solutions provide limited informa-tion on their local behavior The starting point is a sup bound for the solutionsand weak forms of the maximum principle A further step is their local H¨oldercontinuity
An introduction to these local methods is in Chapter 10 in the framework
of DeGiorgi classes While originating from quasi-linear elliptic equations,
Trang 19xviii Preface to the Second Edition
these classes have a life of their own The investigation of the local and ary behavior of functions in these classes, involves a combination of methodsfrom PDEs, measure theory, and harmonic analysis We start by tracing themback to quasi-linear elliptic equations, and then present in detail some ofthese methods In particular, we establish that functions in these classes arelocally bounded and locally H¨older continuous, and we give conditions for theregularity to extend up to the boundary Finally, we prove that non-negativefunctions on the DeGiorgi classes satisfy the Harnack inequality This, on theone hand, is a surprising fact, since these classes require only some sort ofCaccioppoli-type energy bounds On the other hand, this raises the question
bound-of understanding their structure, which to date is still not fully understood.While some facts about these classes are scattered in the literature, this is per-haps the first systematic presentation of DeGiorgi classes in their own right.Some of the material is as recent as last year In this respect, these last twochapters provide a background on a spectrum of techniques in local behavior
of solutions of elliptic PDEs, and build toward research topics of current activeinvestigation
The presentation is more terse and streamlined than in the first tion Some elementary background material (Weierstrass Theorem, mollifiers,Ascoli–Arzel´a Theorem, Jensen’s inequality, etc ) has been removed
edi-I am indebted to many colleagues and students who, over the past fourteenyears, have offered critical suggestions and pointed out misprints, imprecisestatements, and points that were not clear on a first reading Among theseGiovanni Caruso, Xu Guoyi, Hanna Callender, David Petersen, Mike O’Leary,Changyong Zhong, Justin Fitzpatrick, Abey Lopez and Haichao Wang Specialthanks go to Matt Calef for reading carefully a large portion of the manu-script and providing suggestions and some simplifying arguments The help
of U Gianazza has been greatly appreciated He has read the entire script with extreme care and dedication, picking up points that needed to beclarified I am very much indebted to Ugo
manu-I would like to thank Avner Friedman, James Serrin, ConstantineDafermos, Bob Glassey, Giorgio Talenti, Luigi Ambrosio, Juan Manfredi,John Lewis, Vincenzo Vespri, and Gui Qiang Chen for examining the manu-script in detail and for providing valuable comments Special thanks to DavidKinderlehrer for his suggestion to include material on weak formulations anddirect methods Without his input and critical reading, the last two chaptersprobably would not have been written Finally, I would like to thank AnnKostant and the entire team at Birkh¨auser for their patience in coping with
my delays
June 2009
Trang 20These notes are meant to be a self contained, elementary introduction topartial differential equations (PDEs) They assume only advanced differential
calculus and some basic L p theory Although the basic equations treated inthis book, given its scope, are linear, I have made an attempt to approachthen from a non-linear perspective
Chapter I is focused on the Cauchy–Kowalewski theorem We discuss thenotion of characteristic surfaces and use it to classify partial differential equa-tions The discussion grows from equations of second-order in two variables to
equations of second-order in N variables to PDEs of any order in N variables.
In Chapters 2 and 3 we study the Laplace equation and connected tic theory The existence of solutions for the Dirichlet problem is proven bythe Perron method This method clarifies the structure of the sub(super)-
ellip-harmonic functions, and it is closely related to the modern notion of viscosity
solution The elliptic theory is complemented by the Harnack and Liouville
theorems, the simplest version of Schauder’s estimates, and basic L p-potentialestimates Then, in Chapter 3 the Dirichlet and Neumann problems, as well
as eigenvalue problems for the Laplacian, are cast in terms of integral tions This requires some basic facts concerning double-layer potentials and
equa-the notion of compact subsets of L p, which we present
In Chapter 4 we present the Fredholm theory of integral equations andderive necessary and sufficient conditions for solving the Neumann problem
We solve eigenvalue problems for the Laplacian, generate orthonormal systems
in L2, and discuss questions of completeness of such systems in L2 Thisprovides a theoretical basis for the method of separation of variables.Chapter 5 treats the heat equation and related parabolic theory We intro-duce the representation formulas, and discuss various comparison principles.Some focus has been placed on the uniqueness of solutions to the Cauchyproblem and their behavior as |x| → ∞ We discuss Widder’s theorem and
the structure of the non-negative solutions To prove the parabolic Harnackestimate we have used an idea introduced by Krylov and Safonov in the con-text of fully non-linear equations
Trang 21xx Preface to the First Edition
The wave equation is treated in Chapter 6 in its basic aspects We deriverepresentation formulas and discuss the role of the characteristics, propaga-tion of signals, and questions of regularity For general linear second-orderhyperbolic equations in two variables, we introduce the Riemann function andprove its symmetry properties The sections on Goursat problems represent aconcrete application of integral equations of Volterra type
Chapter 7 is an introduction to conservation laws The main points of thetheory are taken from the original papers of Hopf and Lax from the 1950s.Space is given to the minimization process and the meaning of taking the
initial data in the sense of L1 The uniqueness theorem we present is due
to Kruzhkov (1970) We discuss the meaning of viscosity solution vis-`a-visthe notion of sub-solutions and maximum principle for parabolic equations.The theory is complemented by an analysis of the asymptotic behavior, againfollowing Hopf and Lax
Even though the layout is theoretical, I have indicated some of the physicalorigins of PDEs Reference is made to potential theory, similarity solutionsfor the porous medium equation, generalized Riemann problems, etc
I have also attempted to convey the notion of ill-posed problems, mainly
via some examples of Hadamard
Most of the background material, arising along the presentation, has beenstated and proved in the complements Examples include the Ascoli–Arzel`a
theorem, Jensen’s inequality, the characterization of compactness in L p, fiers, basic facts on convex functions, and the Weierstrass theorem A book
molli-of this kind is bound to leave out a number molli-of topics, and this book is noexception Perhaps the most noticeable omission here is some treatment ofnumerical methods
These notes have grown out of courses in PDEs I taught over the years
at Indiana University, Northwestern University and the University of Rome
II, Italy My thanks go to the numerous students who have pointed out prints and imprecise statements Of these, special thanks go to M O’Leary,
mis-D Diller, R Czech, and A Grillo I am indebted to A Devinatz for reading
a large portion of the manuscript and for providing valuable critical ments I have also benefited from the critical input of M Herrero, V Vesprii,and J Manfredi, who have examined parts of the manuscript I am grate-ful to E Giusti for his help with some of the historical notes The input of
com-L Chierchia has been crucial He has read a large part of the manuscriptand made critical remarks and suggestions He has also worked out in detail
a large number of the problems and supplied some of his own In particular,
he wrote the first draft of problems 2.7–2.13 of Chapter 5 and 6.10–6.11 of
Chapter 6 Finally I like to thank M Cangelli and H Howard for their helpwith the graphics
Trang 221 Green’s Theorem
Let E be an open set in RN , and let k be a non-negative integer Denote
by C k (E) the collection of all real-valued, k-times continuously differentiable functions in E A function f is in C k (E) if f ∈ C k (E), and its support
is contained in E A function f : ¯ E → R is in C k( ¯E), if f ∈ C k (E) and all partial derivatives ∂ f /∂x i for all i = 1, , N and = 0, , k, admit continuous extensions up to ∂E The boundary ∂E is of class C1 if for all
y ∈ ∂Ω, there exists ε > 0 such that within the ball B ε (y) centered at y and radius ε, ∂E can be implicitly represented, in a local system of coordinates,
as a level set of a function Φ ∈ C1(B ε (y)) such that |∇Φ| = 0 in B ε (y).
If ∂E is of class C1, let n(x) =
n1(x), , n N (x)
denote the unit normal
exterior to E at x ∈ ∂E Each of the components n j(·) is well defined as a
continuous function on ∂E A real vector-valued function
¯
E x → f(x) =f1(x), , f N (x)
∈ R N
is of class C k (E), C k( ¯E), or C k (E) if all components f jbelong to these classes
Theorem 1.1 Let E be a bounded domain of RN with boundary ∂E of class
C1 Then for every f ∈ C1( ¯E)
This is also referred to as the divergence theorem, or as the formula of
inte-gration by parts It continues to hold if n is only dσ-a.e defined in ∂E For
example, ∂E could be a cube inRN More generally, ∂E could be the finite
E DiBenedetto, Partial Differential Equations: Second Edition,
Cornerstones, DOI 10.1007/978-0-8176-4552-6_1,
© Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2010
1
Trang 232 0 Preliminaries
union of portions of surfaces of class C1 The domain E need not be bounded,
provided|f| and |∇f| decay sufficiently fast as |x| → ∞.1
1.1 Differential Operators and Adjoints
Given a symmetric matrix (a ij) ∈ R N × R N, a vector b ∈ R N , and c ∈ R,
consider the formal expression
ThusL = L ∗ if b = 0 If u, v ∈ C2( ¯E) for a bounded open set E ⊂ R N with
boundary ∂E of class C1, the divergence theorem yields the Green’s formula
More generally, the entries of the matrix (a ij ) as well as b and c might be
smooth functions of x In such a case, for v ∈ C2( ¯E), define
o (E) If u and v do not vanish near ∂E, a version of (1.2) continues to
hold, where the right-hand side contains the extra boundary integral
Identifying precise conditions on ∂E and f for which one can integrate by parts
is part of geometric measure theory ([56]).
Trang 242 The Continuity Equation
Let t → E(t) be a set-valued function that associates to each t in some open
interval I ⊂ R a bounded open set E(t) ⊂ R N , for some N ≥ 2 Assume
that the boundaries ∂E(t) are uniformly of class C1, and that there exists a
bounded open set E ⊂ R N such that E(t) ⊂ E, for all t ∈ I Our aim is to
compute the derivative
d
dt
E(t)
ρ(x, t)dx for a given ρ ∈ C1(E × I).
Regard points x ∈ E(t) as moving along the trajectories t → x(t) with
veloci-ties ˙x = v(x, t) Assume that the motion, or deformation, of E( ·) is smooth
in the sense that (x, t) → v(x, t) is continuous in a neighborhood of E × I.
Forming the difference quotient gives
As for the second, first compute the difference of the last two volume integrals
by means of Riemann sums as follows Fix a number 0 < Δσ 1, and
approximate ∂E(t) by means of a polyhedron with faces of area not exceeding
Δσ and tangent to ∂E(t) at some of their interior points Let {F1, , F n }
for some n ∈ N be a finite collection of faces making up the approximating
polyhedron, and let x i for i = 1, , n, be a selection of their tangency points with ∂E(t) Then approximate the set
E(t + Δt) − E(t)
E(t) − E(t + Δt)
by the union of the cylinders of basis F i and height v(x i , t) · nΔt, built with
their axes parallel to the outward normal to ∂E(t) at x i Therefore, for Δt
fixed
Trang 25Consider now an ideal fluid filling a region E ⊂ R3 Assume that the fluid is
compressible (say a gas) and let (x, t) → ρ(x, t) denote its density At some
instant t, cut a region E(t) out of E and follow the motion of E(t) as if each of
its points were identified with the moving particles Whatever the sub-region
E(t), during the motion the mass is conserved Therefore
d dt
Trang 263 The Heat Equation and the Laplace Equation
Any quantity that is conserved as it moves within an open set E with velocity
v satisfies the conservation law (2.4) Let u be the temperature of a material
homogeneous body occupying the region E If c is the heat capacity, the thermal energy stored at x ∈ E at time t is cu(x, t) By Fourier’s law the
energy “moves” following gradients of temperature, i.e.,
Now let u be the pressure of a fluid moving with velocity v through a region
E ofRN occupied by a porous medium The porosity p oof the medium is therelative infinitesimal fraction of space occupied by the pores and available to
the fluid Let μ, k, and ρ denote respectively kinematic viscosity, permeability,
and density By Darcy’s law ([137])
v =− kp o
Assume that k and μ are constant If the fluid is incompressible, then ρ =
const, and it follows from (2.4) that div v = 0 Therefore the pressure u
satisfies
div∇u = Δu = u x i x i = 0 in E. (3.4)
The latter is the Laplace equation for the function u A fluid whose velocity
is given as the gradient of a scalar function is a potential fluid ([160]).
3.1 Variable Coefficients
Consider now the same physical phenomena taking place in non-homogeneous,anisotropic media For heat conduction in such media, temperature gradientsmight generate heat propagation in preferred directions, which themselves
might depend on x ∈ E As an example one might consider the heat diffusion
in a solid of given conductivity, in which is embedded a bundle of curvilinearmaterial fibers of different conductivity Thus in general, the conductivity
of the composite medium is a tensor dependent on the location x ∈ E and
time t, represented formally by an N × N matrix k = k ij (x, t)
For such
a tensor, the product on the right-hand side of (3.1) is the row-by-column
product of the matrix (k ij) and the column vector∇u Enforcing the same
conservation of energy (2.4) yields a non-homogeneous, anisotropic version ofthe heat equation (3.2), in the form
u t −a ij (x, t)u x i
x j = 0 in E, where a ij= k ij
Trang 276 0 Preliminaries
Similarly, the permeability of a non-homogeneous, anisotropic porous medium
is a position-independent tensor
k ij (x) Then, analogous considerations
applied to (3.3), imply that the velocity potential u of the flow of a fluid in
a heterogeneous, anisotropic porous medium satisfies the partial differentialequation
The physical, tensorial nature of either heat conductivity or permeability of a
medium implies that (a ij ) is symmetric, bounded, and positive definite in E.
However, no further information is available on these coefficients, since theyreflect interior properties of physical domains, not accessible without alteringthe physical phenomenon we are modeling This raises the question of the
meaning of (3.5)–(3.6) Indeed, even if u ∈ C2(E), the indicated operations are not well defined for a ij ∈ L ∞ (E) A notion of solution will be given in
Chapter 9, along with solvability methods
Equations (3.5)–(3.6) are said to be in divergence form Equations in divergence form are of the type
and arise in the theory of stochastic control ([89])
4 A Model for the Vibrating String
Consider a material string of constant linear density ρ whose end points are fixed, say at 0 and 1 Assume that the string is vibrating in the plane (x, y), set the interval (0, 1) on the x-axis, and let (x, t) → u(x, t) be the y-coordinate
of the string at the point x ∈ (0, 1) at the instant t ∈ R The basic physical
assumptions are:
(i) The dimensions of the cross sections are negligible with respect to the
length, so that the string can be identified, for all t, with the graph of
x → u(x, t).
(ii) Let (x, t) → T(x, t) denote the tension, i.e., the sum of the internal forces
per unit length, generated by the displacement of the string Assume that
T at each point (x, u(x, t)) is tangent to the string Letting T = |T|,
assume that (x, t) → T (x, t) is t-independent.
(iii) Resistance of the material to flexure is negligible with respect to thetension
(iv) Vibrations are small in the sense that |u| θ and |u x | θ for θ > 1 are
negligible when compared with|u| and |u x |.
Trang 28Consider next, for t fixed, a small interval (x1, x2)⊂ (0, 1) and the
correspond-ing portion of the strcorrespond-ing of extremities
p(x, t)dx, where p( ·, t) = {load per unit length}.
3 The inertial forces due to the vertical acceleration u tt (x, t), i.e.,
Trang 29for some T o > 0 independent of x In view of the physical assumptions (ii)
and (iv), may take T o also independent of t These remarks in (4.1) yield the
partial differential equation
This is the wave equation in one space variable.
Remark 4.1 The assumption that ρ is constant is a “linear” assumption
in the sense that leads to the linear wave equation (4.2) Non-linear effectsdue to variable density were already observed by D Bernoulli ([11]), and byS.D Poisson ([120])
5 Small Vibrations of a Membrane
A membrane is a rigid thin body of constant density ρ, whose thickness is
negligible with respect to its extension Assume that, at rest, the membrane
occupies a bounded open set E ⊂ R2, and that it begins to vibrate under the
action of a vertical load, say (x, t) → p(x, t) Identify the membrane with the
graph of a smooth function (x, t) → u(x, t) defined in E × R and denote by
∇u = (u x1, u x2) the spatial gradient of u The relevant physical assumptions
are:
Trang 30(i) Forces due to flexure are negligible.
(ii) Vibrations occur only in the direction u normal to the position of rest of
the membrane Moreover, vibrations are small, in the sense that u x i u x j
and uu x i for i, j = 1, 2 are negligible when compared to u and |∇u|.
(iii) The tension T has constant modulus, say|T| = T o > 0.
Cut a small ideal region G o ⊂ E with boundary ∂G o of class C1, and let G
be the corresponding portion of the membrane Thus G is the graph of u( ·, t)
restricted to G o , or equivalently, G o is the projection on the plane u = 0 of the portion G of the membrane Analogously, introduce the curve Γ limiting
G and its projection Γ o = ∂G o The tension T acts at points P ∈ Γ and is
tangent to G at P and normal to Γ If τ is the unit vector of T and n is the
exterior unit normal to G at P , let e be the unit tangent to Γ at P oriented
so that the triple{τ, e, n} is positive and τ = e ∧ n Our aim is to compute
the vertical component of T at P ∈ Γ If {i, j, k} is the positive unit triple
along the coordinate axes, we will compute the quantity T· k = T o τ · k.
Consider a parametrization of Γ o, say
s → P o (s) =
x1(s), x2(s)
for s ∈ {some interval of R}.
The unit exterior normal to ∂G o is given by
ν = (x 2, −x
1)
Trang 31Next, write down the equation of instantaneous equilibrium of the portion G
of the membrane The vertical loads on G, the vertical contribution of the
tension T, and the inertial force due to acceleration u tt are respectively
Trang 32Therefore
G o
[ρu tt − T odiv∇u − p] dx = 0
for all t ∈ R and all G o ⊂ E Thus
u tt − c2Δu = f in E × R (5.1)where
c2=T o
ρ , f =
p
ρ , Δu = div( ∇u).
Equation (5.1), modeling small vibrations of a stretched membrane, is thetwo-dimensional wave equation
6 Transmission of Sound Waves
An ideal compressible fluid is moving within a region E ⊂ R3 Let ρ(x, t) and
v(x, t) denote its density and velocity at x ∈ E at the instant t Each x can
be regarded as being in motion along the trajectory t → x(t) with velocity
where∇ denotes the gradient with respect to the space variables only Cut any
region G o ⊂ E with boundary ∂G o of class C1 Since G o is instantaneously
in equilibrium, the balance of forces acting on G o must be zero These are:(i) The inertial forces due to acceleration
G o
ρ
vt+ (v· ∇)vdx.
(ii) The Kelvin forces due to pressure Let p(x, t) be the pressure at x ∈ E
at time t The forces due to pressure on G are
∂G0
p ν dσ, ν = {outward unit normal to ∂G o }.
(iii) The sum of the external forces, and the internal forces due to friction
−
G
f dx.
Trang 33(a) The fluid moves with small relative velocity and small time variations
of density Therefore second-order terms of the type v i v j,x h and ρ t v i arenegligible with respect to first order terms
(b) Heat transfer is slower than pressure drops, i.e., the process is adiabatic
and ρ = h(p) for some h ∈ C2(R)
Expanding h( ·) about the equilibrium pressure p o, renormalized to be zero,gives
ρ = a o p + a1p2+· · ·
Assume further that the pressure is close to the equilibrium pressure, so that
all terms of order higher than one are negligible when compared to a o p These
Combining these remarks gives the equation of the pressure in the propagation
of sound waves in a fluid, in the form
∂2p
∂t2 − c2Δp = f in E × R (6.2)where
Trang 347 The Navier–Stokes System
The system (6.1) is rather general and holds for any ideal fluid If the fluid is
incompressible, then ρ = const, and the continuity equation (2.4) gives
If in addition the fluid is viscous, the internal forces due to friction can be
represented by μρΔv, where μ > 0 is the kinematic viscosity ([160]) Therefore
(6.1) yields the system
∂
∂tv− μΔv + (v · ∇)v + 1
ρ ∇p = f e (7.2)
where fe = f /ρ are the external forces acting on the system The unknowns
are the three components of the velocity and the pressure p, to be determined
from the system of four equations (7.1) and (7.2)
8 The Euler Equations
Let S denote the entropy function of a gas undergoing an adiabatic process The pressure p and the density ρ are linked by the equation of state
Trang 3514 0 Preliminaries
9 Isentropic Potential Flows
A flow is isentropic if S = const In this case, the equation of state (8.1)
permits one to define the pressure as a function of the density alone Let
u ∈ C2(R3×R) be the velocity potential, so that v = ∇u Assume that f = 0,
has the dimension of the square of a velocity, and c represents the local speed
of sound Notice that c need not be constant Next multiply the ith equation
in (9.1) by u x i and add for i = 1, 2, 3 to obtain
12
∂
∂t |∇u|2+1
2∇u · ∇|∇u|2=−1
ρ ∇p · ∇u. (9.3)Using the continuity equation
Trang 36From the equation of state
d
dt f (S) =
d dt
p
ρ 1+α = 1
ρ α
d dt
p
ρ+
p ρ
d dt
9.1 Steady Potential Isentropic Flows
For steady flows, rewrite (9.4) in the form
λ1= 1− |∇u|2
c2 and λ2= 1.
Using the steady-state version of the Bernoulli law (9.2) gives the first
eigen-value in terms only of the pressure p and the density ρ The ratio M = |∇u|/c
of the speed of a body to the speed of sound in the surrounding medium iscalled the Mach number.3
3Ernst Mach, 1838–1916 Mach one is the speed of sound; Mach two is twice thespeed of sound;
Trang 3716 0 Preliminaries
10 Partial Differential Equations
The equations and systems of the previous sections are examples of PDEs
Let u ∈ C m (E) for some m ∈ N, and for j = 1, 2, , m, let D j u denote
the vector of all the derivatives of u of order j For example, if N = m = 2, denoting (x, y) the coordinates inR2
The PDE is of order m if the gradient of F with respect to D m u is not
identically zero It is linear if for all u, v ∈ C m (E) and all α, β ∈ R
It is quasi-linear if it is linear with respect to the highest order derivatives.
Typically a quasi-linear PDE takes the form
where m j are non-negative integers and the coefficients a m1, ,m N, and the
forcing term F o , are given smooth functions of (x, u, D1u, D2u, , D m −1 u).
If the PDE is quasi-linear, the sum of the terms involving the derivatives ofhighest order, is the principal part of the PDE
Trang 38Quasi-Linear Equations and the
E DiBenedetto, Partial Differential Equations: Second Edition,
© Birkhäuser Boston, a part of Springer Science + Business Media, LLC 2010
Cornerstones, DOI 10.1007/978-0-8176-4552-6_2, 17
Trang 3918 1 Quasi-Linear Equations and the Cauchy–Kowalewski Theorem
Here A, B, C are known, since they are computed on Γ Precisely
Using (1.4) and the matrix M , we classify, locally, the family of quasi-linear
equations (1.1) as elliptic if det M > 0, i.e., if there exists no real teristic; parabolic if det M = 0, i.e., if there exists one family of real char- acteristics; hyperbolic if det M < 0, i.e., if there exist two families of real
charac-characteristics The elliptic, parabolic, or hyperbolic nature of (1.1) may bedifferent in different regions ofR2 For example, the Tricomi equation ([152])
yu xx − u yy= 0
is elliptic in the region [y < 0], parabolic on the x-axis and hyperbolic in the upper half-plane [y > 0] The characteristics are solutions of √
yy = ±1 in
the upper half-plane [y > 0].
The elliptic, parabolic, or hyperbolic nature of the PDE may also dependupon the solution itself As an example, consider the equation of steady com-
pressible fluid flow of a gas of density u and velocity ∇u = (u x , u y) inR2,introduced in (9.5) of the Preliminaries
(c2− u2
x )u xx − 2u x u y u xy + (c2− u2
y )u yy= 0
Trang 40where c > 0 is the speed of sound Compute
Therefore the equation is elliptic for sub-sonic flow (|∇u| < c), parabolic for
sonic flow (|∇u| = c), and hyperbolic for super-sonic flow (|∇u| > c) The
is hyperbolic with characteristic lines x ± cy = const.
2 Characteristics and Singularities
If Γ is a characteristic, the Cauchy problem (1.1)–(1.2) is in general not able, since the second derivatives of u cannot be computed on Γ We may attempt to solve the PDE (1.1) on each side of Γ and then piece together the functions so obtained Assume that Γ dividesR2 into two regions E1and E2and let u i ∈ C2( ¯E i ), for i = 1, 2, be possible solutions of (1.1) in E isatisfyingthe Cauchy data (1.2) These are taken in the sense of
f =
f1 in E1
f2 in E2let [f ] denote the jump of f across Γ , i.e.,
[f ](t) = lim
(x,y)→(ξ(t),η(t)) (x,y)∈E1
f1(x, y) − lim
(x,y)→(ξ(t),η(t)) (x,y)∈E2
f2(x, y).