Elliptic & Parabolic Equations

Elliptic & Parabolic Equations Zhuoqun Wu, Jingxue Yin & Chunpeng Wang Jilin University, China World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI

Contents Preface v 1. Preliminary Knowledge 1 1.1 Some Frequently Applied Inequalities and Basic Techniques 1 1.1.1 Some frequently applied inequalities 1 1.1.2 Spaces C fc (Q) and C#(f2) 2 1.1.3 Smoothing operators 3 1.1.4 Cut-off functions 5 1.1.5 Partition of unity 6 1.1.6 Local flatting of the boundary 6 1.2 Holder Spaces 7 1.2.1 Spaces C k < a {Ti) and C k ' a {Sl) 7 1.2.2 Interpolation inequalities 8 1.2.3 Spaces C 2k+a ' k+a / 2 (Q T ) 13-1.3 Isotropic Sobolev Spaces 14 1.3.1 Weak derivatives 14 1.3.2 Sobolev spaces W k ' p (Q) and w ' p (tt) 15 1.3.3 Operation rules of weak derivatives 17 1.3.4 Interpolation inequality 17 1.3.5 Embedding theorem 19 1.3.6 Poincare's inequality 21 1.4 <-Anisotropic Sobolev Spaces 24 1.4.1 Spaces W 2k ' k (Q T ), W^'HQT), W 2 p k ' k (Q T ), V 2 {QT) and V(Q T ) 24 1.4.2 Embedding theorem 26 1.4.3 Poincare's inequality 28

x Elliptic and Parabolic Equations 1.5 Trace of Functions in H^Q) 29 1.5.1 Some propositions on functions in H 1 (Q + ) 29 1.5.2 Trace of functions in H 1^) 33 1.5.3 Trace of functions in H\Q T ) = Wl' x {Q T ) 35 2. L 2 Theory of Linear Elliptic Equations 39 2.1 Weak Solutions of Poisson's Equation 39 2.1.1 Definition of weak solutions 40 2.1.2 Riesz's representation theorem and its application.. 41 2.1.3 Transformation of the problem 43 2.1.4 Existence of minimizers of the corresponding functional 44 2.2 Regularity of Weak Solutions of Poisson's Equation... 47 2.2.1 Difference operators 47 2.2.2 Interior regularity 50 2.2.3 Regularity near the boundary 53 2.2.4 Global regularity 56 2.2.5 Study of regularity by means of smoothing operators 58 2.3 L 2 Theory of General Elliptic Equations 60 2.3.1 Weak solutions 60 2.3.2 Riesz's representation theorem and its application.. 61 2.3.3 Variational method 62 2.3.4 Lax-Milgram's theorem and its application 64 2.3.5 Fredholm's alternative theorem and its application. 67 3. L 2 Theory of Linear Parabolic Equations 71 3.1 Energy Method 71 3.1.1 Definition of weak solutions 72 3.1.2 A modified Lax-Milgram's theorem 73 3.1.3 Existence and uniqueness of the weak solution.... 75 3.2 Rothe's Method 79 3.3 Galerkin's Method 85 3.4 Regularity of Weak Solutions 89 3.5 L 2 Theory of General Parabolic Equations 94 3.5.1 Energy method 94 3.5.2 Rothe's method 96 3.5.3 Galerkin's method 97 4. De Giorgi Iteration and Moser Iteration 105

Contents xi 4.1 Global Boundedness Estimates of Weak Solutions of Poisson's Equation 105.:.. 4.1.1 Weak maximum principle for solutions of Laplace's equation 105 4.1.2 Weak maximum principle for solutions of Poisson's equation 107 4.2 Global Boundedness Estimates for Weak Solutions of the Heat Equation Ill 4.2.1 Weak maximum principle for solutions of the homogeneous heat equation Ill 4.2.2 Weak maximum principle for solutions of the nonhomogeneous heat equation 112 4.3 Local Boundedness Estimates for Weak Solutions of Poisson's Equation 116 4.3.1 Weak subsolutions (supersolutions) 116 4.3.2 Local boundedness estimate for weak solutions of Laplace's equation 118 4.3.3 Local boundedness estimate for solutions of Poisson's equation 120 4.3.4 Estimate near the boundary for weak solutions of Poisson's equation 122 4.4 Local Boundedness Estimates for Weak Solutions of the Heat Equation 123 4.4.1 Weak subsolutions (supersolutions) 123 4.4.2 Local boundedness estimate for weak solutions of the homogeneous heat equation 123 4.4.3 Local boundedness estimate for weak solutions of the nonhomogeneous heat equation 126 5. Harnack's Inequalities 131 5.1 Harnack's Inequalities for Solutions of Laplace's Equation. 131 5.1.1 Mean value formula 131 5.1.2 Classical Harnack's inequality 133 5.1.3 Estimate of sup u 133 BBR 5.1.4 Estimate of inf u 135 BBR 5.1.5 Harnack's inequality 141 5.1.6 Holder's estimate 143

xii Elliptic and Parabolic Equations 5.2 Harnack's Inequalities for Solutions of the Homogeneous Heat Equation 145 5.2.1 Weak Harnack's inequality 146 5.2.2 Holder's estimate 155 5.2.3 Harnack's inequality 156 6. Schauder's Estimates for Linear Elliptic Equations 159 6.1 Campanato Spaces 159 6.2 Schauder's Estimates for Poisson's Equation 165 6.2.1 Estimates to be established 165 6.2.2 Caccioppoli's inequalities 168 6.2.3 Interior estimate for Laplace's equation 173 6.2.4 Near boundary estimate for Laplace's equation... 175 6.2.5 Iteration lemma 177 6.2.6 Interior estimate for Poisson's equation 178 6.2.7 Near boundary estimate for Poisson's equation... 181 6.3 Schauder's Estimates for General Linear Elliptic Equations 187 6.3.1 Simplification of the problem 188 6.3.2 Interior estimate 188 6.3.3 Near boundary estimate 191 6.3.4 Global estimate 193 7. Schauder's Estimates for Linear Parabolic Equations 197 7.1 i-anisotropic Campanato Spaces 197 7.2 Schauder's Estimates for the Heat Equation 199 7.2.1 Estimates to be established 199 7.2.2 Interior estimate 200 7.2.3 Near bottom estimate 208 7.2.4 Near lateral estimate 214 7.2.5 Near lateral-bottom estimate 227 7.2.6 Schauder's estimates for general linear parabolic equations 231 8. Existence of Classical Solutions for Linear Equations 233 8.1 Maximum Principle and Comparison Principle 233 8.1.1 The case of elliptic equations 233 8.1.2 The case of parabolic equations 236

Contents xiii 8.2 Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations 240 8.2.1 Existence and uniqueness of the classical solution for Poisson's equation 240 8.2.2 The method of continuity 246 8.2.3 Existence and uniqueness of classical solutions for general linear elliptic equations 248 8.3 Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations 249 8.3.1 Existence and uniqueness of the classical solution for the heat equation 250 8.3.2 Existence and uniqueness of classical solutions for general linear parabolic equations 251 9. L p Estimates for Linear Equations and Existence of Strong Solutions 255 9.1 IP Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions 255 9.1.1 L p estimates for Poisson's equation in cubes 255 9.1.2 L p estimates for general linear elliptic equations... 260 9.1.3 Existence and uniqueness of strong solutions for linear elliptic equations 264 9.2 V Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions 266 9.2.1 L p estimates for the heat equation in cubes 266 9.2.2 L p estimates for general linear parabolic equations. 271 9.2.3 Existence and uniqueness of strong solutions for linear parabolic equations 272 10. Fixed Point Method 277 10.1 Framework of Solving Quasilinear Equations via Fixed Point Method 277 10.1.1 Leray-Schauder's fixed point theorem 277 10.1.2 Solvability of quasilinear elliptic equations 277 10.1.3 Solvability of quasilinear parabolic equations 280 10.1.4 The procedures of the a priori estimates 282 10.2 Maximum Estimate 282 10.3 Interior Holder's Estimate 284

xiv Elliptic and Parabolic Equations 10.4 Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation 287 10.5 Boundary Holder's Estimate and Boundary Gradient Estimate 289 10.6 Global Gradient Estimate 296 10.7 Holder's Estimate for a Linear Equation 301 10.7.1 An iteration lemma 301 10.7.2 Morrey's theorem 302 10.7.3 Holder's estimate 303 10.8 Holder's Estimate for Gradients 307 10.8.1 Interior Holder's estimate for gradients of solutions. 307 10.8.2 Boundary Holder's estimate for gradients of solutions 308 10.8.3 Global Holder's estimate for gradients of solutions. 310 10.9 Solvability of More General Quasilinear Equations 310 10.9.1 Solvability of more general quasilinear elliptic equations 310 10.9.2 Solvability of more general quasilinear parabolic equations 311 11. Topological Degree Method 313 11.1 Topological Degree 313 11.1.1 Brouwer degree 313 11.1.2 Leray-Schauder degree 315 11.2 Existence of a Heat Equation with Strong Nonlinear Source 317 12. Monotone Method 323 12.1 Monotone Method for Parabolic Problems 323 12.1.1 Definition of supersolutions and subsolutions 324 12.1.2 Iteration and monotone property 324 12.1.3 Existence results 327 12.1.4 Application to more general parabolic equations... 330 12.1.5 Nonuniqueness of solutions 332 12.2 Monotone Method for Coupled Parabolic Systems 336 12.2.1 Quasimonotone reaction functions 337 12.2.2 Definition of supersolutions and subsolutions 337 12.2.3 Monotone sequences 339 12.2.4 Existence results 350 12.2.5 Extension 353

Contents xv 13. Degenerate Equations 355 13.1 Linear Equations 355 13.1.1 Formulation of the first boundary value problem.. 356 13.1.2 Solvability of the problem in a space similar to H 1. 361 13.1.3 Solvability of the problem in L p (Cl)... 362 13.1.4 Method of elliptic regularization 365 13.1.5 Uniqueness of weak solutions in L p (fl) and regularity 366 13.2 A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations 368 13.2.1 Definition of weak solutions 369 13.2.2 Uniqueness of weak solutions for one dimensional equations 371 13.2.3 Existence of weak solutions for one dimensional equations 373 13.2.4 Uniqueness of weak solutions for higher dimensional equations 378 13.2.5 Existence of weak solutions for higher dimensional equations 381 13.3 General Quasilinear Degenerate Parabolic Equations... 384 13.3.1 Uniqueness of weak solutions for weakly degenerate equations 385 13.3.2 Existence of weak solutions for weakly degenerate equations 393 13.3.3 A remark on quasilinear parabolic equations with strong degeneracy 399 Bibliography 403 Index 405