Regularity Estimates for Nonlinear Elliptic and Parabolic Problems
Regularity Estimates for Nonlinear Elliptic and Parabolic Problems

Author: John Lewis

Publisher: Springer Science & Business Media

Published: 2012-03-02

Total Pages: 259

ISBN-13: 3642271448

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The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions. Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.

Regularity Estimates for Nonlinear Elliptic and Parabolic Problems
Regularity Estimates for Nonlinear Elliptic and Parabolic Problems

Author: John Lewis

Publisher: Springer

Published: 2012-03-02

Total Pages: 247

ISBN-13: 9783642271458

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The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions. Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.

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.

Nonlinear Elliptic and Parabolic Problems
Nonlinear Elliptic and Parabolic Problems

Author: Michel Chipot

Publisher: Springer Science & Business Media

Published: 2006-02-09

Total Pages: 531

ISBN-13: 3764373857

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Celebrates the work of the renowned mathematician Herbert Amann, who had a significant and decisive influence in shaping Nonlinear Analysis. Containing 32 contributions, this volume covers a range of nonlinear elliptic and parabolic equations, with applications to natural sciences and engineering.

Elliptic and Parabolic Problems in Non Smooth Domains
Elliptic and Parabolic Problems in Non Smooth Domains

Author: Ian Wood

Publisher: Logos Verlag Berlin

Published: 2005

Total Pages: 0

ISBN-13: 9783832510596

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Regularity of solutions is an important part of the theory of partial differential equations. In this text, the regularity of solutions to elliptic and parabolic problems in Lipschitz domains is investigated. Maximal regularity estimates are useful when dealing with nonlinear parabolic problems. However, the known maximal regularity results for smooth domains no longer hold in Lp-spaces over Lipschitz domains for the whole range of exponents p. Here, maximal regularity estimates are shown for the Laplacian with suitable domain in Lp-spaces for a restricted range of p. Operators with L-coefficients in convex domains and Ornstein-Uhlenbeck operators in exterior Lipschitz domains are also discussed.

Nonlinear Elliptic and Parabolic Equations of the Second Order
Nonlinear Elliptic and Parabolic Equations of the Second Order

Author: N.V. Krylov

Publisher: Springer

Published: 2001-11-30

Total Pages: 0

ISBN-13: 9781402003349

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Approach your problems from the It isn't that they can't see the right end and begin with the solution. It is that they can't see answers. Then one day, perhaps the problem. you will find the final question. G.K. Chesterton. The Scandal of 'The Hermit Clad in Crane Father Brown 'The Point of a Pin'. Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono graphs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theor.etical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Elliptic and Parabolic Problems
Elliptic and Parabolic Problems

Author: Catherine Bandle

Publisher: Springer Science & Business Media

Published: 2006-01-17

Total Pages: 466

ISBN-13: 3764373849

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Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.

Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations
Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations

Author: N. V. Krylov

Publisher: American Mathematical Soc.

Published: 2018-09-07

Total Pages: 458

ISBN-13: 1470447401

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This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov–Safonov and the Evans–Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman–Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cut-off” equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions.

Maximal L_1tnp-L_1tnq Regularity for Discrete Elliptic Differential Operators
Maximal L_1tnp-L_1tnq Regularity for Discrete Elliptic Differential Operators

Author: Matthias Geißert

Publisher:

Published: 2003

Total Pages: 127

ISBN-13: 9783936846515

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Maximal regularity estimates play an important role in the analysis of parabolic problems. In particular, maximal regularity estimates are very useful for nonlinear parabolic problems. Therefore, in this book, maximal regularity estimates for approximations of solutions of parabolic differential equations by finite elements or finite differences are investigated. In this context, one has to show that maximal regularity estimates are independent of the mesh. More precisely, uniform estimates for a family of finite element (finite difference) operators are shown. These estimates are used to prove uniqueness and existence of a local, strong solution for approximations of semilinear problems. Moreover, uniform maximal regularity estimates lead to a new approach for proving error estimates. Finally, error estimates in the linear and semilinear case are proved.

Elliptic & Parabolic Equations
Elliptic & Parabolic Equations

Author: Zhuoqun Wu

Publisher: World Scientific

Published: 2006

Total Pages: 428

ISBN-13: 9812700250

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This book provides an introduction to elliptic and parabolic equations. While there are numerous monographs focusing separately on each kind of equations, there are very few books treating these two kinds of equations in combination. This book presents the related basic theories and methods to enable readers to appreciate the commonalities between these two kinds of equations as well as contrast the similarities and differences between them.