Elliptic & parabolic equations /

Elliptic & parabolic equations /

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Bibliographic Details
Author / Creator:Wu, Zhuoqun.
Imprint:Singapore ; Hackensack, NJ : World Scientific, c2006.
Description:xv, 408 p. ; 24 cm.
Language:English
Subject:
Differential equations, Elliptic.
Differential equations, Parabolic.
Differential equations, Elliptic.
Differential equations, Parabolic.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/6377788
Hidden Bibliographic Details
Varying Form of Title:Elliptic and parabolic equations
Other authors / contributors:Yin, Jingxue.
Wang, Chunpeng.
ISBN:9812700250
9789812700254
9812700269 (pbk)
9789812700261 (pbk)
Notes:Includes bibliographical references (p. 403-404) and index.
Table of Contents:
  • Preface
  • 1. Preliminary Knowledge
  • 1.1. Some Frequently Applied Inequalities and Basic Techniques
  • 1.1.1. Some frequently applied inequalities
  • 1.1.2. Spaces C[superscript k]([Omega]) and C[Characters not reproducible]([Omega])
  • 1.1.3. Smoothing operators
  • 1.1.4. Cut-off functions
  • 1.1.5. Partition of unity
  • 1.1.6. Local flatting of the boundary
  • 1.2. Holder Spaces
  • 1.2.1. Spaces C[superscript k, alpha]([Characters not reproducible]) and C[superscript k, alpha]([Omega])
  • 1.2.2. Interpolation inequalities
  • 1.2.3. Spaces C[superscript 2k+alpha,k+alpha/2]([Characters not reproducible][subscript T])
  • 1.3. Isotropic Sobolev Spaces
  • 1.3.1. Weak derivatives
  • 1.3.2. Sobolev spaces W[superscript k,p]([Omega]) and W[Characters not reproducible]([Omega])
  • 1.3.3. Operation rules of weak derivatives
  • 1.3.4. Interpolation inequality
  • 1.3.5. Embedding theorem
  • 1.3.6. Poincare's inequality
  • 1.4. t-Anisotropic Sobolev Spaces
  • 1.4.1. Spaces W[Characters not reproducible] (Q[subscript T]), W[Characters not reproducible](Q[subscript T]), W[Characters not reproducible]), V[subscript 2](Q[subscript T]) and V(Q[subscript T])
  • 1.4.2. Embedding theorem
  • 1.4.3. Poincare's inequality
  • 1.5. Trace of Functions in H[superscript 1]([Omega])
  • 1.5.1. Some propositions on functions in H[superscript 1](Q[superscript +])
  • 1.5.2. Trace of functions in H[superscript 1]([Omega])
  • 1.5.3. Trace of functions in H[superscript 1](Q[subscript T]) = W[Characters not reproducible](Q[superscript T])
  • 2. L[superscript 2] Theory of Linear Elliptic Equations
  • 2.1. Weak Solutions of Poisson's Equation
  • 2.1.1. Definition of weak solutions
  • 2.1.2. Riesz's representation theorem and its application
  • 2.1.3. Transformation of the problem
  • 2.1.4. Existence of minimizers of the corresponding functional
  • 2.2. Regularity of Weak Solutions of Poisson's Equation
  • 2.2.1. Difference operators
  • 2.2.2. Interior regularity
  • 2.2.3. Regularity near the boundary
  • 2.2.4. Global regularity
  • 2.2.5. Study of regularity by means of smoothing operators
  • 2.3. L[superscript 2] Theory of General Elliptic Equations
  • 2.3.1. Weak solutions
  • 2.3.2. Riesz's representation theorem and its application
  • 2.3.3. Variational method
  • 2.3.4. Lax-Milgram's theorem and its application
  • 2.3.5. Fredholm's alternative theorem and its application
  • 3. L[superscript 2] Theory of Linear Parabolic Equations
  • 3.1. Energy Method
  • 3.1.1. Definition of weak solutions
  • 3.1.2. A modified Lax-Milgram's theorem
  • 3.1.3. Existence and uniqueness of the weak solution
  • 3.2. Rothe's Method
  • 3.3. Galerkin's Method
  • 3.4. Regularity of Weak Solutions
  • 3.5. L[superscript 2] Theory of General Parabolic Equations
  • 3.5.1. Energy method
  • 3.5.2. Rothe's method
  • 3.5.3. Galerkin's method
  • 4. De Giorgi Iteration and Moser Iteration
  • 4.1. Global Boundedness Estimates of Weak Solutions of Poisson's Equation
  • 4.1.1. Weak maximum principle for solutions of Laplace's equation
  • 4.1.2. Weak maximum principle for solutions of Poisson's equation
  • 4.2. Global Boundedness Estimates for Weak Solutions of the Heat Equation
  • 4.2.1. Weak maximum principle for solutions of the homogeneous heat equation
  • 4.2.2. Weak maximum principle for solutions of the nonhomogeneous heat equation
  • 4.3. Local Boundedness Estimates for Weak Solutions of Poisson's Equation
  • 4.3.1. Weak subsolutions (supersolutions)
  • 4.3.2. Local boundedness estimate for weak solutions of Laplace's equation
  • 4.3.3. Local boundedness estimate for solutions of Poisson's equation
  • 4.3.4. Estimate near the boundary for weak solutions of Poisson's equation
  • 4.4. Local Boundedness Estimates for Weak Solutions of the Heat Equation
  • 4.4.1. Weak subsolutions (supersolutions)
  • 4.4.2. Local boundedness estimate for weak solutions of the homogeneous heat equation
  • 4.4.3. Local boundedness estimate for weak solutions of the nonhomogeneous heat equation
  • 5. Harnack's Inequalities
  • 5.1. Harnack's Inequalities for Solutions of Laplace's Equation
  • 5.1.1. Mean value formula
  • 5.1.2. Classical Harnack's inequality
  • 5.1.3. Estimate of [Characters not reproducible] u
  • 5.1.4. Estimate of [Characters not reproducible] u
  • 5.1.5. Harnack's inequality
  • 5.1.6. Holder's estimate
  • 5.2. Harnack's Inequalities for Solutions of the Homogeneous Heat Equation
  • 5.2.1. Weak Harnack's inequality
  • 5.2.2. Holder's estimate
  • 5.2.3. Harnack's inequality
  • 6. Schauder's Estimates for Linear Elliptic Equations
  • 6.1. Campanato Spaces
  • 6.2. Schauder's Estimates for Poisson's Equation
  • 6.2.1. Estimates to be established
  • 6.2.2. Caccioppoli's inequalities
  • 6.2.3. Interior estimate for Laplace's equation
  • 6.2.4. Near boundary estimate for Laplace's equation
  • 6.2.5. Iteration lemma
  • 6.2.6. Interior estimate for Poisson's equation
  • 6.2.7. Near boundary estimate for Poisson's equation
  • 6.3. Schauder's Estimates for General Linear Elliptic Equations
  • 6.3.1. Simplification of the problem
  • 6.3.2. Interior estimate
  • 6.3.3. Near boundary estimate
  • 6.3.4. Global estimate
  • 7. Schauder's Estimates for Linear Parabolic Equations
  • 7.1. t-Anisotropic Campanato Spaces
  • 7.2. Schauder's Estimates for the Heat Equation
  • 7.2.1. Estimates to be established
  • 7.2.2. Interior estimate
  • 7.2.3. Near bottom estimate
  • 7.2.4. Near lateral estimate
  • 7.2.5. Near lateral-bottom estimate
  • 7.2.6. Schauder's estimates for general linear parabolic equations
  • 8. Existence of Classical Solutions for Linear Equations
  • 8.1. Maximum Principle and Comparison Principle
  • 8.1.1. The case of elliptic equations
  • 8.1.2. The case of parabolic equations
  • 8.2. Existence and Uniqueness of Classical Solutions for Linear Elliptic Equations
  • 8.2.1. Existence and uniqueness of the classical solution for Poisson's equation
  • 8.2.2. The method of continuity
  • 8.2.3. Existence and uniqueness of classical solutions for general linear elliptic equations
  • 8.3. Existence and Uniqueness of Classical Solutions for Linear Parabolic Equations
  • 8.3.1. Existence and uniqueness of the classical solution for the heat equation
  • 8.3.2. Existence and uniqueness of classical solutions for general linear parabolic equations
  • 9. L[superscript p] Estimates for Linear Equations and Existence of Strong Solutions
  • 9.1. L[superscript p] Estimates for Linear Elliptic Equations and Existence and Uniqueness of Strong Solutions
  • 9.1.1. L[superscript p] estimates for Poisson's equation in cubes
  • 9.1.2. L[superscript p] estimates for general linear elliptic equations
  • 9.1.3. Existence and uniqueness of strong solutions for linear elliptic equations
  • 9.2. L[superscript p] Estimates for Linear Parabolic Equations and Existence and Uniqueness of Strong Solutions
  • 9.2.1. L[superscript p] estimates for the heat equation in cubes
  • 9.2.2. L[superscript p] estimates for general linear parabolic equations
  • 9.2.3. Existence and uniqueness of strong solutions for linear parabolic equations
  • 10. Fixed Point Method
  • 10.1. Framework of Solving Quasilinear Equations via Fixed Point Method
  • 10.1.1. Leray-Schauder's fixed point theorem
  • 10.1.2. Solvability of quasilinear elliptic equations
  • 10.1.3. Solvability of quasilinear parabolic equations
  • 10.1.4. The procedures of the a priori estimates
  • 10.2. Maximum Estimate
  • 10.3. Interior Holder's Estimate
  • 10.4. Boundary Holder's Estimate and Boundary Gradient Estimate for Solutions of Poisson's Equation
  • 10.5. Boundary Holder's Estimate and Boundary Gradient Estimate
  • 10.6. Global Gradient Estimate
  • 10.7. Holder's Estimate for a Linear Equation
  • 10.7.1. An iteration lemma
  • 10.7.2. Morrey's theorem
  • 10.7.3. Holder's estimate
  • 10.8. Holder's Estimate for Gradients
  • 10.8.1. Interior Holder's estimate for gradients of solutions
  • 10.8.2. Boundary Holder's estimate for gradients of solutions
  • 10.8.3. Global Holder's estimate for gradients of solutions
  • 10.9. Solvability of More General Quasilinear Equations
  • 10.9.1. Solvability of more general quasilinear elliptic equations
  • 10.9.2. Solvability of more general quasilinear parabolic equations
  • 11. Topological Degree Method
  • 11.1. Topological Degree
  • 11.1.1. Brouwer degree
  • 11.1.2. Leray-Schauder degree
  • 11.2. Existence of a Heat Equation with Strong Nonlinear Source
  • 12. Monotone Method
  • 12.1. Monotone Method for Parabolic Problems
  • 12.1.1. Definition of supersolutions and subsolutions
  • 12.1.2. Iteration and monotone property
  • 12.1.3. Existence results
  • 12.1.4. Application to more general parabolic equations
  • 12.1.5. Nonuniqueness of solutions
  • 12.2. Monotone Method for Coupled Parabolic Systems
  • 12.2.1. Quasimonotone reaction functions
  • 12.2.2. Definition of supersolutions and subsolutions
  • 12.2.3. Monotone sequences
  • 12.2.4. Existence results
  • 12.2.5. Extension
  • 13. Degenerate Equations
  • 13.1. Linear Equations
  • 13.1.1. Formulation of the first boundary value problem
  • 13.1.2. Solvability of the problem in a space similar to H[superscript 1]
  • 13.1.3. Solvability of the problem in L[superscript p]([Omega])
  • 13.1.4. Method of elliptic regularization
  • 13.1.5. Uniqueness of weak solutions in L[superscript p]([Omega]) and regularity
  • 13.2. A Class of Special Quasilinear Degenerate Parabolic Equations - Filtration Equations
  • 13.2.1. Definition of weak solutions
  • 13.2.2. Uniqueness of weak solutions for one dimensional equations
  • 13.2.3. Existence of weak solutions for one dimensional equations
  • 13.2.4. Uniqueness of weak solutions for higher dimensional equations
  • 13.2.5. Existence of weak solutions for higher dimensional equations
  • 13.3. General Quasilinear Degenerate Parabolic Equations
  • 13.3.1. Uniqueness of weak solutions for weakly degenerate equations
  • 13.3.2. Existence of weak solutions for weakly degenerate equations
  • 13.3.3. A remark on quasilinear parabolic equations with strong degeneracy
  • Bibliography
  • Index
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