Stable Solutions of Elliptic Partial Differential Equations

Stable Solutions of Elliptic Partial Differential Equations PDF

Author: Louis Dupaigne

Publisher: CRC Press

Published: 2011-03-15

Total Pages: 334

ISBN-13: 1420066552

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Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Stable Solutions of Elliptic Partial Differential Equations

Stable Solutions of Elliptic Partial Differential Equations PDF

Author: Louis Dupaigne

Publisher: CRC Press

Published: 2011-03-15

Total Pages: 337

ISBN-13: 1420066544

DOWNLOAD EBOOK →

Stable solutions are ubiquitous in differential equations. They represent meaningful solutions from a physical point of view and appear in many applications, including mathematical physics (combustion, phase transition theory) and geometry (minimal surfaces). Stable Solutions of Elliptic Partial Differential Equations offers a self-contained presentation of the notion of stability in elliptic partial differential equations (PDEs). The central questions of regularity and classification of stable solutions are treated at length. Specialists will find a summary of the most recent developments of the theory, such as nonlocal and higher-order equations. For beginners, the book walks you through the fine versions of the maximum principle, the standard regularity theory for linear elliptic equations, and the fundamental functional inequalities commonly used in this field. The text also includes two additional topics: the inverse-square potential and some background material on submanifolds of Euclidean space.

Fine Regularity of Solutions of Elliptic Partial Differential Equations

Fine Regularity of Solutions of Elliptic Partial Differential Equations PDF

Author: Jan Malý

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 309

ISBN-13: 0821803352

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The primary objective of this monograph is to give a comprehensive exposition of results surrounding the work of the authors concerning boundary regularity of weak solutions of second order elliptic quasilinear equations in divergence form. The book also contains a complete development of regularity of solutions of variational inequalities, including the double obstacle problem, where the obstacles are allowed to be discontinuous. The book concludes with a chapter devoted to the existence theory thus providing the reader with a complete treatment of the subject ranging from regularity of weak solutions to the existence of weak solutions.

Elliptic Partial Differential Equations

Elliptic Partial Differential Equations PDF

Author: Lucio Boccardo

Publisher: Walter de Gruyter

Published: 2013-10-29

Total Pages: 204

ISBN-13: 3110315424

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Elliptic partial differential equations is one of the main and most active areas in mathematics. This book is devoted to the study of linear and nonlinear elliptic problems in divergence form, with the aim of providing classical results, as well as more recent developments about distributional solutions. For this reason this monograph is addressed to master's students, PhD students and anyone who wants to begin research in this mathematical field.

Degenerate Elliptic Equations

Degenerate Elliptic Equations PDF

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.

Elliptic Partial Differential Equations

Elliptic Partial Differential Equations PDF

Author: Qing Han

Publisher: American Mathematical Soc.

Published: 2011

Total Pages: 161

ISBN-13: 0821853139

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This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems.

Stability of Solutions of Integrable Partial Differential Equations

Stability of Solutions of Integrable Partial Differential Equations PDF

Author: Jeremy Upsal

Publisher:

Published: 2020

Total Pages: 101

ISBN-13:

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Stability analysis for solutions of partial differential equations (PDEs) is important for determining the applicability of a model to the physical world. Establishing stability for PDE solutions is often significantly more challenging than for ordinary differential equation solutions. This task becomes tractable for PDEs possessing a Lax pair. In this dissertation, I provide a general framework for computing large parts of the Lax spectrum for periodic and quasiperiodic solutions of a general class of PDEs possessing a Lax pair. This class consists of the AKNS hierarchy admitting a common reduction and generalizations. I then relate the Lax spectrum to the stability spectrum using the squared-eigenfunction connection. Using this, I demonstrate that the subset of the real line which is part of the Lax spectrum maps to stable elements of the linearization. Several examples that demonstrate the direct applicability of this work are provided. One example is worked out in detail: the stability analysis for the elliptic solutions of the focusing nonlinear Schrödinger (NLS) equation. For the NLS equation, I go further by establishing orbital stability of the elliptic solutions with respect to a class of perturbations of integer multiples of the period of the solution.

Elliptic Partial Differential Equations of Second Order

Elliptic Partial Differential Equations of Second Order PDF

Author: David Gilbarg

Publisher: Springer Science & Business Media

Published: 2001-01-12

Total Pages: 544

ISBN-13: 9783540411604

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This work aims to be of interest to those who have to work with differential equations and acts either as a reference or as a book to learn from. The authors have made the treatment self-contained.

Finite Difference Methods for Ordinary and Partial Differential Equations

Finite Difference Methods for Ordinary and Partial Differential Equations PDF

Author: Randall J. LeVeque

Publisher: SIAM

Published: 2007-01-01

Total Pages: 356

ISBN-13: 9780898717839

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This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples.

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