Second Order Parabolic Differential Equations
World Scientific, 1996 - 439 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.
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INTRODUCTION TO THE THEORY OF WEAK
EXISTENCE UNIQUENESS AND REGULARITY
FURTHER THEORY OF WEAK SOLUTIONS
FIXED POINT THEOREMS AND THEIR APPLICATIONS
COMPARISON AND MAXIMUM PRINCIPLES
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analog argument assume boundary condition boundary data boundary gradient estimate boundary value Cauchy-Dirichlet problem Chapter coefficients conormal problem constant C determined convex Corollary cylinder define denote differential equation Dirichlet Dirichlet problem divergence form eigenvalues elliptic equations existence result existence theorem finite follows fully nonlinear gradient bound Harnack inequality hence Holder continuous Holder estimate holds implies infer integral linear Lipschitz Math matrix maximum principle mean curvature mean curvature equation modulus of continuity Monge-Ampere equation nonnegative constant norm oblique derivative problem parabolic equations pointwise positive constants proof of Theorem Proposition prove quasilinear quasilinear equations regularity satisfies second derivative Section shows simple Sobolev inequality solvability space structure conditions subset sufficiently small suitable Suppose test function Theorem 6.1 theory tions uniformly parabolic equations unique solution Ural'tseva vector weak derivatives weak Harnack inequality weak solution write
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