Second Order Parabolic Differential EquationsWorld Scientific, 6 nov 1996 - 452 páginas This book is an introduction to the general theory of second order parabolic differential equations, which model many important, time-dependent physical systems. It studies the existence, uniqueness, and regularity of solutions to a variety of problems with Dirichlet boundary conditions and general linear and nonlinear boundary conditions by means of a priori estimates. The first seven chapters give a description of the linear theory and are suitable for a graduate course on partial differential equations. The last eight chapters cover the nonlinear theory for smooth solutions. They include much of the author's research and are aimed at researchers in the field. A unique feature is the emphasis on time-varying domains. |
Índice
1 | |
7 | |
CHAPTER III INTRODUCTION TO THE THEORY OF WEAK SOLUTIONS | 21 |
CHAPTER IV HÖLDER ESTIMATES | 45 |
CHAPTER V EXISTENCE UNIQUENESS AND REGULARITYOF SOLUTIONS | 87 |
CHAPTER VI FURTHER THEORY OF WEAK SOLUTIONS | 101 |
CHAPTER VII STRONG SOLUTIONS | 155 |
CHAPTER VIII FIXED POINT THEOREMS AND THEIR APPLICATIONS | 203 |
CHAPTER X BOUNDARY GRADIENT ESTIMATES | 231 |
CHAPTER XI GLOBAL AND LOCAL GRADIENT BOUNDS | 259 |
CHAPTER X HÖLDER GRADIENT ESTIMATES AND EXISTENCE THEOREMS | 301 |
CHAPTER XIII THE OBLIQUE DERIVATIVE PROBLEM FOR QUASILINEAR PARABOLIC EQUATIONS | 321 |
CHAPTER XIV FULLY NONLINEAR EQUATIONS IINTRODUCTION | 361 |
CHAPTER XV FULLY NONLINEAR EQUATIONS II HESSIAN EQUATIONS | 385 |
Bibliography | 421 |
445 | |
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argument assume boundary condition boundary gradient estimate boundary value problem Cauchy-Dirichlet problem Chapter coefficients constant C determined convex Corollary cylinder define Dirichlet problem divergence form domain Dºu Duſ elliptic equations English transl equa existence result existence theorem f in Q finite fixed follows fully nonlinear global gradient bound H2+o Harnack inequality hence Holder estimate Holder gradient holds hypotheses implies infer integral Let u e linear Lipschitz Math matrix maximum principle mean curvature modulus of continuity Monge-Ampere equation nonnegative constant norm oblique derivative problem parabolic equations Partial Differential Equations pointwise positive constants proof of Theorem Proposition prove quasilinear regularity respect Russian satisfies second derivative Section shows simple solvability space subset subsolution sufficiently small suitable Suppose Theorem 6.1 tions unique solution vector weak derivatives weak Harnack inequality weak solution write