Elliptic & Parabolic Equations

Transcription

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

2 Contents Preface v 1. Preliminary Knowledge Some Frequently Applied Inequalities and Basic Techniques Some frequently applied inequalities Spaces C fc (Q) and C#(f2) Smoothing operators Cut-off functions Partition of unity Local flatting of the boundary Holder Spaces Spaces C k < a {Ti) and C k ' a {Sl) Interpolation inequalities Spaces C 2k+a ' k+a / 2 (Q T ) Isotropic Sobolev Spaces Weak derivatives Sobolev spaces W k ' p (Q) and w ' p (tt) Operation rules of weak derivatives Interpolation inequality Embedding theorem Poincare's inequality <-Anisotropic Sobolev Spaces Spaces W 2k ' k (Q T ), W^'HQT), W 2 p k ' k (Q T ), V 2 {QT) and V(Q T ) Embedding theorem Poincare's inequality 28

3 x Elliptic and Parabolic Equations 1.5 Trace of Functions in H^Q) Some propositions on functions in H 1 (Q + ) Trace of functions in H 1^) Trace of functions in H\Q T ) = Wl' x {Q T ) L 2 Theory of Linear Elliptic Equations Weak Solutions of Poisson's Equation Definition of weak solutions Riesz's representation theorem and its application Transformation of the problem Existence of minimizers of the corresponding functional Regularity of Weak Solutions of Poisson's Equation Difference operators Interior regularity Regularity near the boundary Global regularity Study of regularity by means of smoothing operators L 2 Theory of General Elliptic Equations Weak solutions Riesz's representation theorem and its application Variational method Lax-Milgram's theorem and its application Fredholm's alternative theorem and its application L 2 Theory of Linear Parabolic Equations Energy Method Definition of weak solutions A modified Lax-Milgram's theorem Existence and uniqueness of the weak solution Rothe's Method Galerkin's Method Regularity of Weak Solutions L 2 Theory of General Parabolic Equations Energy method Rothe's method Galerkin's method De Giorgi Iteration and Moser Iteration 105

4 Contents xi 4.1 Global Boundedness Estimates of Weak Solutions of Poisson's Equation 105.: Weak maximum principle for solutions of Laplace's equation Weak maximum principle for solutions of Poisson's equation Global Boundedness Estimates for Weak Solutions of the Heat Equation Ill Weak maximum principle for solutions of the homogeneous heat equation Ill Weak maximum principle for solutions of the nonhomogeneous heat equation Local Boundedness Estimates for Weak Solutions of Poisson's Equation Weak subsolutions (supersolutions) Local boundedness estimate for weak solutions of Laplace's equation Local boundedness estimate for solutions of Poisson's equation Estimate near the boundary for weak solutions of Poisson's equation Local Boundedness Estimates for Weak Solutions of the Heat Equation Weak subsolutions (supersolutions) Local boundedness estimate for weak solutions of the homogeneous heat equation Local boundedness estimate for weak solutions of the nonhomogeneous heat equation Harnack's Inequalities Harnack's Inequalities for Solutions of Laplace's Equation Mean value formula Classical Harnack's inequality Estimate of sup u 133 BBR Estimate of inf u 135 BBR Harnack's inequality Holder's estimate 143

5 xii Elliptic and Parabolic Equations 5.2 Harnack's Inequalities for Solutions of the Homogeneous Heat Equation Weak Harnack's inequality Holder's estimate Harnack's inequality Schauder's Estimates for Linear Elliptic Equations Campanato Spaces Schauder's Estimates for Poisson's Equation Estimates to be established Caccioppoli's inequalities Interior estimate for Laplace's equation Near boundary estimate for Laplace's equation Iteration lemma Interior estimate for Poisson's equation Near boundary estimate for Poisson's equation Schauder's Estimates for General Linear Elliptic Equations Simplification of the problem Interior estimate Near boundary estimate Global estimate Schauder's Estimates for Linear Parabolic Equations i-anisotropic Campanato Spaces Schauder's Estimates for the Heat Equation Estimates to be established Interior estimate Near bottom estimate Near lateral estimate Near lateral-bottom estimate Schauder's estimates for general linear parabolic equations Existence of Classical Solutions for Linear Equations Maximum Principle and Comparison Principle The case of elliptic equations The case of parabolic equations 236

6 Contents xiii 8.2 Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations Existence and uniqueness of the classical solution for Poisson's equation The method of continuity Existence and uniqueness of classical solutions for general linear elliptic equations Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations Existence and uniqueness of the classical solution for the heat equation Existence and uniqueness of classical solutions for general linear parabolic equations L p Estimates for Linear Equations and Existence of Strong Solutions IP Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions L p estimates for Poisson's equation in cubes L p estimates for general linear elliptic equations Existence and uniqueness of strong solutions for linear elliptic equations V Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions L p estimates for the heat equation in cubes L p estimates for general linear parabolic equations Existence and uniqueness of strong solutions for linear parabolic equations Fixed Point Method Framework of Solving Quasilinear Equations via Fixed Point Method Leray-Schauder's fixed point theorem Solvability of quasilinear elliptic equations Solvability of quasilinear parabolic equations The procedures of the a priori estimates Maximum Estimate Interior Holder's Estimate 284

7 xiv Elliptic and Parabolic Equations 10.4 Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation Boundary Holder's Estimate and Boundary Gradient Estimate Global Gradient Estimate Holder's Estimate for a Linear Equation An iteration lemma Morrey's theorem Holder's estimate Holder's Estimate for Gradients Interior Holder's estimate for gradients of solutions Boundary Holder's estimate for gradients of solutions Global Holder's estimate for gradients of solutions Solvability of More General Quasilinear Equations Solvability of more general quasilinear elliptic equations Solvability of more general quasilinear parabolic equations Topological Degree Method Topological Degree Brouwer degree Leray-Schauder degree Existence of a Heat Equation with Strong Nonlinear Source Monotone Method Monotone Method for Parabolic Problems Definition of supersolutions and subsolutions Iteration and monotone property Existence results Application to more general parabolic equations Nonuniqueness of solutions Monotone Method for Coupled Parabolic Systems Quasimonotone reaction functions Definition of supersolutions and subsolutions Monotone sequences Existence results Extension 353

8 Contents xv 13. Degenerate Equations Linear Equations Formulation of the first boundary value problem Solvability of the problem in a space similar to H Solvability of the problem in L p (Cl) Method of elliptic regularization Uniqueness of weak solutions in L p (fl) and regularity A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations Definition of weak solutions Uniqueness of weak solutions for one dimensional equations Existence of weak solutions for one dimensional equations Uniqueness of weak solutions for higher dimensional equations Existence of weak solutions for higher dimensional equations General Quasilinear Degenerate Parabolic Equations Uniqueness of weak solutions for weakly degenerate equations Existence of weak solutions for weakly degenerate equations A remark on quasilinear parabolic equations with strong degeneracy 399 Bibliography 403 Index 405