A Priori Estimate and Some Properties of Solutions of Elliptic and Parabolic Equations
A Priori Estimate and Some Properties of Solutions of Elliptic and Parabolic Equations

Author: S. N. Kruzhkov

Publisher:

Published: 1968

Total Pages: 62

ISBN-13:

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In this paper we establish a priori estimates and costudy certain qualitative properties of generalized solutions of second-order elliptic and parabolic equations. The first fundamental results in this direction for equations with many independent variables were established in the papers by de Giorgi and Nash. A refined new approach to the consideration of similar problems and, in particular, a new proof of the de Giorgio theorem were given by Moser. In the present paper the method proposed by Moser is expanded.

Second Order Equations of Elliptic and Parabolic Type
Second Order Equations of Elliptic and Parabolic Type

Author: E. M. Landis

Publisher: American Mathematical Soc.

Published: 1997-12-02

Total Pages: 224

ISBN-13: 9780821897812

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Most books on elliptic and parabolic equations emphasize existence and uniqueness of solutions. By contrast, this book focuses on the qualitative properties of solutions. In addition to the discussion of classical results for equations with smooth coefficients (Schauder estimates and the solvability of the Dirichlet problem for elliptic equations; the Dirichlet problem for the heat equation), the book describes properties of solutions to second order elliptic and parabolic equations with measurable coefficients near the boundary and at infinity. The book presents a fine elementary introduction to the theory of elliptic and parabolic equations of second order. The precise and clear exposition is suitable for graduate students as well as for research mathematicians who want to get acquainted with this area of the theory of partial differential equations.

Qualitative Theory of Parabolic Equations, Part 1
Qualitative Theory of Parabolic Equations, Part 1

Author: T. I. Zelenyak

Publisher: Walter de Gruyter

Published: 2011-09-06

Total Pages: 425

ISBN-13: 311093504X

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In the qualitative theory of ordinary differential equations, the Liapunov method plays a fundamental role. To use their analogs for the analysis of stability of solutions to parabolic, hyperparabolic, and other nonclassical equations and systems, time-invariant a priori estimates have to be devised for solutions. In this publication only parabolic problems are considered. Here lie, mainly, the problems which have been investigated most thoroughly --- the construction of Liapunov functionals which naturally generalize Liapunov functions for nonlinear parabolic equations of the second order with one spatial variable. The authors establish stabilizing solutions theorems, and the necessary and sufficient conditions of general and asymptotic stability of stationary solutions, including the so-called critical case. Attraction domains for stable solutions of mixed problems for these equations are described. Furthermore, estimates for the number of stationary solutions are obtained.

Singular Solutions of Nonlinear Elliptic and Parabolic Equations
Singular Solutions of Nonlinear Elliptic and Parabolic Equations

Author: Alexander A. Kovalevsky

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2016-03-21

Total Pages: 448

ISBN-13: 3110332248

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This monograph looks at several trends in the investigation of singular solutions of nonlinear elliptic and parabolic equations. It discusses results on the existence and properties of weak and entropy solutions for elliptic second-order equations and some classes of fourth-order equations with L1-data and questions on the removability of singularities of solutions to elliptic and parabolic second-order equations in divergence form. It looks at localized and nonlocalized singularly peaking boundary regimes for different classes of quasilinear parabolic second- and high-order equations in divergence form. The book will be useful for researchers and post-graduate students that specialize in the field of the theory of partial differential equations and nonlinear analysis. Contents: Foreword Part I: Nonlinear elliptic equations with L^1-data Nonlinear elliptic equations of the second order with L^1-data Nonlinear equations of the fourth order with strengthened coercivity and L^1-data Part II: Removability of singularities of the solutions of quasilinear elliptic and parabolic equations of the second order Removability of singularities of the solutions of quasilinear elliptic equations Removability of singularities of the solutions of quasilinear parabolic equations Quasilinear elliptic equations with coefficients from the Kato class Part III: Boundary regimes with peaking for quasilinear parabolic equations Energy methods for the investigation of localized regimes with peaking for parabolic second-order equations Method of functional inequalities in peaking regimes for parabolic equations of higher orders Nonlocalized regimes with singular peaking Appendix: Formulations and proofs of the auxiliary results Bibliography

Lectures on Elliptic and Parabolic Equations in Holder Spaces
Lectures on Elliptic and Parabolic Equations in Holder Spaces

Author: Nikolaĭ Vladimirovich Krylov

Publisher: American Mathematical Soc.

Published: 1996

Total Pages: 178

ISBN-13: 082180569X

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These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in H older spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, containing 200 exercises, aims to provide a good understanding of what kind of results are available and what kinds of techniques are used to obtain them.

Partial Differential Equations of Elliptic Type
Partial Differential Equations of Elliptic Type

Author: C. Miranda

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 384

ISBN-13: 3642877737

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In the theory of partial differential equations, the study of elliptic equations occupies a preeminent position, both because of the importance which it assumes for various questions in mathematical physics, and because of the completeness of the results obtained up to the present time. In spite of this, even in the more classical treatises on analysis the theory of elliptic equations has been considered and illustrated only from particular points of view, while the only expositions of the whole theory, the extremely valuable ones by LICHTENSTEIN and AscoLI, have the charac ter of encyclopedia articles and date back to many years ago. Consequently it seemed to me that it would be of some interest to try to give an up-to-date picture of the present state of research in this area in a monograph which, without attaining the dimensions of a treatise, would nevertheless be sufficiently extensive to allow the expo sition, in some cases in summary form, of the various techniques used in the study of these equations.

Elliptic and Parabolic Equations Involving the Hardy-Leray Potential
Elliptic and Parabolic Equations Involving the Hardy-Leray Potential

Author: Ireneo Peral Alonso

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2021-02-22

Total Pages: 514

ISBN-13: 3110606275

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The scientific literature on the Hardy-Leray inequality, also known as the uncertainty principle, is very extensive and scattered. The Hardy-Leray potential shows an extreme spectral behavior and a peculiar influence on diffusion problems, both stationary and evolutionary. In this book, a big part of the scattered knowledge about these different behaviors is collected in a unified and comprehensive presentation.