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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addition analog apply argument assume boundary boundary condition calculation Chapter choose coefficients consider continuous Corollary corresponding curvature cylinder define definition denote depending derivative determined differential equation domain example Exercise existence extended Finally fixed follows function further given gives global gradient bound gradient estimate H2+a hence Hölder estimate holds hypotheses implies inequality infer integral interior Lemma linear matrix maximum principle mean method modified Moreover nonlinear nonnegative norm oblique derivative problem obtain operator parabolic equations particular positive constants problem PROOF Proposition prove quasilinear regularity replaced respect result satisfies shows similar simple smooth solves space step structure subset sufficiently suitable Suppose Theorem theory tions uniformly unique vector weak solution write
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