Existence and Regularity of Generalized Solutions of Degenerate Parabolic Problems Via Finite Difference Numerical Schemes
Existence and Regularity of Generalized Solutions of Degenerate Parabolic Problems Via Finite Difference Numerical Schemes

Author: Rasoul Behboudi

Publisher:

Published: 2002

Total Pages: 100

ISBN-13:

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ABSTRACT: This work is focused on questions of solvability of initial-boundary value problems for degenerate parabolic partial differential equations, and on the investigation of differential and certain qualitative properties of solutions of such equations. The generalized solvability of initial-boundary value problems is proved for linear equations with a degeneracy of parabolicity at the boundaries by employing numerical solutions obtained from finite difference numerical schemes. Furthermore, the regularity of generalized solutions is studied.

Degenerate Parabolic Equations
Degenerate Parabolic Equations

Author: Emmanuele DiBenedetto

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 402

ISBN-13: 1461208955

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Evolved from the author's lectures at the University of Bonn's Institut für angewandte Mathematik, this book reviews recent progress toward understanding of the local structure of solutions of degenerate and singular parabolic partial differential equations.

Degenerate Elliptic Equations
Degenerate Elliptic Equations

Author: Serge Levendorskii

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 442

ISBN-13: 9401712158

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This volume is the first to be devoted to the study of various properties of wide classes of degenerate elliptic operators of arbitrary order and pseudo-differential operators with multiple characteristics. Conditions for operators to be Fredholm in appropriate weighted Sobolev spaces are given, a priori estimates of solutions are derived, inequalities of the Grding type are proved, and the principal term of the spectral asymptotics for self-adjoint operators is computed. A generalization of the classical Weyl formula is proposed. Some results are new, even for operators of the second order. In addition, an analogue of the Boutet de Monvel calculus is developed and the index is computed. For postgraduate and research mathematicians, physicists and engineers whose work involves the solution of partial differential equations.

Partial Differential Equations in China
Partial Differential Equations in China

Author: Chaohao Gu

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 193

ISBN-13: 9401111987

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In the past few years there has been a fruitful exchange of expertise on the subject of partial differential equations (PDEs) between mathematicians from the People's Republic of China and the rest of the world. The goal of this collection of papers is to summarize and introduce the historical progress of the development of PDEs in China from the 1950s to the 1980s. The results presented here were mainly published before the 1980s, but, having been printed in the Chinese language, have not reached the wider audience they deserve. Topics covered include, among others, nonlinear hyperbolic equations, nonlinear elliptic equations, nonlinear parabolic equations, mixed equations, free boundary problems, minimal surfaces in Riemannian manifolds, microlocal analysis and solitons. For mathematicians and physicists interested in the historical development of PDEs in the People's Republic of China.

Fully Nonlinear Elliptic Equations
Fully Nonlinear Elliptic Equations

Author: Luis A. Caffarelli

Publisher: American Mathematical Soc.

Published: 1995

Total Pages: 114

ISBN-13: 0821804375

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The goal of the book is to extend classical regularity theorems for solutions of linear elliptic partial differential equations to the context of fully nonlinear elliptic equations. This class of equations often arises in control theory, optimization, and other applications. The authors give a detailed presentation of all the necessary techniques. Instead of treating these techniques in their greatest generality, they outline the key ideas and prove the results needed for developing the subsequent theory. Topics discussed in the book include the theory of viscosity solutions for nonlinear equations, the Alexandroff estimate and Krylov-Safonov Harnack-type inequality for viscosity solutions, uniqueness theory for viscosity solutions, Evans and Krylov regularity theory for convex fully nonlinear equations, and regularity theory for fully nonlinear equations with variable coefficients.

Strongly Coupled Parabolic and Elliptic Systems
Strongly Coupled Parabolic and Elliptic Systems

Author: Dung Le

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2018-11-05

Total Pages: 198

ISBN-13: 3110608766

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Strongly coupled (or cross-diffusion) systems of parabolic and elliptic partial differential equations appear in many physical applications. This book presents a new approach to the solvability of general strongly coupled systems, a much more difficult problem in contrast to the scalar case, by unifying, elucidating and extending breakthrough results obtained by the author, and providing solutions to many open fundamental questions in the theory. Several examples in mathematical biology and ecology are also included. Contents Interpolation Gagliardo–Nirenberg inequalities The parabolic systems The elliptic systems Cross-diffusion systems of porous media type Nontrivial steady-state solutions The duality RBMO(μ)–H1(μ)| Some algebraic inequalities Partial regularity