Elliptic and Parabolic Equations

The following sections are included:

• Preface

• Contents

The following sections are included:

• Some Frequently Applied Inequalities and Basic Techniques
  • Some frequently applied inequalities
  • Spaces C^k(Ω) and
  • Smoothing operators
  • Cut-off functions
  • Partition of unity
  • Local flatting of the boundary

• Hölder Spaces
  • Spaces and C^k,α(Ω)
  • Interpolation inequalities
  • Spaces

• Isotropic Sobolev Spaces
  • Weak derivatives
  • Sobolev spaces W^k,P(Ω) and
  • Operation rules of weak derivatives
  • Interpolation inequality
  • Embedding theorem
  • Poincaré's inequality

• t-Anisotropic Sobolev Spaces
  • Spaces , , , V₂(Q_T) and V(Q_T)
  • Embedding theorem
  • Poincaré's inequality

• Trace of Functions in H¹(Ω)
  • Some propositions on functions in H¹(Q⁺)
  • Trace of functions in H¹(Ω)
  • Trace of functions in

• Exercises

The following sections are included:

• 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

• L² 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

• Exercises

The following sections are included:

• 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

• L² Theory of General Parabolic Equations
  • Energy method
  • Rothe's method
  • Galerkin's method

• Exercises

The following sections are included:

• Global Boundedness Estimates of Weak Solutions of Poisson's Equation
  • 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
  • Weak maximum principle for solutions of the homogeneous heat equation
  • 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

• Exercises

The following sections are included:

• Harnack's Inequalities for Solutions of Laplace's Equation
  • Mean value formula
  • Classical Harnack's inequality
  • Estimate of
  • Estimate of
  • Harnack's inequality
  • Hölder's estimate

• Harnack's Inequalities for Solutions of the Homogeneous Heat Equation
  • Weak Harnack's inequality
  • Hölder's estimate
  • Harnack's inequality

• Exercises

The following sections are included:

• 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

• Exercises

The following sections are included:

• t-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

• Exercises

The following sections are included:

• Maximum Principle and Comparison Principle
  • The case of elliptic equations
  • The case of parabolic equations

• 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

• Exercises

The following sections are included:

• L^P 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

• L^P 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

• Exercises

The following sections are included:

• 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 Hölder's Estimate

• Boundary Hölder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation

• Boundary Hölder's Estimate and Boundary Gradient Estimate

• Global Gradient Estimate

• Hölder's Estimate for a Linear Equation
  • An iteration lemma
  • Morrey's theorem
  • Hölder's estimate

• Hölder's Estimate for Gradients
  • Interior Hölder's estimate for gradients of solutions
  • Boundary Hölder's estimate for gradients of solutions
  • Global Hölder'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

• Exercises

The following sections are included:

• Topological Degree
  • Brouwer degree
  • Leray-Schauder degree

• Existence of a Heat Equation with Strong Nonlinear Source

• Exercises

The following sections are included:

• 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

• Exercises

The following sections are included:

• Linear Equations
  • Formulation of the first boundary value problem
  • Solvability of the problem in a space similar to H¹
  • Solvability of the problem in L^p(Ω)
  • Method of elliptic regularization
  • Uniqueness of weak solutions in L^p(Ω) 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

The following sections are included:

• Bibliography

• Index