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Author: Louis Dupaigne Publisher: CRC Press ISBN: 1420066552 Category : Mathematics Languages : en Pages : 334
Book Description
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.
Author: Louis Dupaigne Publisher: CRC Press ISBN: 1420066552 Category : Mathematics Languages : en Pages : 334
Book Description
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.
Author: Louis Dupaigne Publisher: CRC Press ISBN: 1420066544 Category : Mathematics Languages : en Pages : 337
Book Description
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.
Author: Jan Malý Publisher: American Mathematical Soc. ISBN: 0821803352 Category : Mathematics Languages : en Pages : 309
Book Description
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.
Author: Lucio Boccardo Publisher: Walter de Gruyter ISBN: 3110315424 Category : Mathematics Languages : en Pages : 204
Book Description
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.
Author: Qing Han Publisher: American Mathematical Soc. ISBN: 0821853139 Category : Mathematics Languages : en Pages : 161
Book Description
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.
Author: Serge Levendorskii Publisher: Springer Science & Business Media ISBN: 9401712158 Category : Mathematics Languages : en Pages : 442
Book Description
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.
Author: Jeremy Upsal Publisher: ISBN: Category : Languages : en Pages : 101
Book Description
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.
Author: Randall J. LeVeque Publisher: SIAM ISBN: 9780898717839 Category : Mathematics Languages : en Pages : 356
Book Description
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.
Author: David Gilbarg Publisher: Springer Science & Business Media ISBN: 9783540411604 Category : Mathematics Languages : en Pages : 544
Book Description
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.
Author: Owe Axelsson Publisher: Bentham Science Publishers ISBN: 1608052915 Category : Mathematics Languages : en Pages : 153
Book Description
This e-book presents several research areas of elliptical problems solved by differential equations. The mathematical models explained in this e-book have been contributed by experts in the field and can be applied to a wide range of real life examples. M
Author: Louis Dupaigne
Author: Louis Dupaigne
Author: Louis Dupaigne
Author: Jan Malý
Author: Lucio Boccardo
Author: Qing Han
Author: Serge Levendorskii
Author: Jeremy Upsal
Author: Randall J. LeVeque
Author: David Gilbarg
Author: Owe Axelsson