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Author : Treven Parker Wall
Publisher : Unknown
Page : 106 pages
File Size : 54,8 Mb
Release : 2007
Category : Electronic
ISBN : CORNELL:31924107954178
Author : A. V. Ivanov
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 45,6 Mb
Release : 1984
Category : Mathematics
ISBN : 0821830805
Author : Lucio Boccardo,Gisella Croce
Publisher : Walter de Gruyter
Page : 201 pages
File Size : 50,5 Mb
Release : 2013-10-29
Category : Mathematics
ISBN : 9783110315424
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 : Bernard Dacorogna,Paolo Marcellini
Publisher : Springer Science & Business Media
Page : 278 pages
File Size : 46,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461215622
Nonlinear partial differential equations has become one of the main tools of mod ern mathematical analysis; in spite of seemingly contradictory terminology, the subject of nonlinear differential equations finds its origins in the theory of linear differential equations, and a large part of functional analysis derived its inspiration from the study of linear pdes. In recent years, several mathematicians have investigated nonlinear equations, particularly those of the second order, both linear and nonlinear and either in divergence or nondivergence form. Quasilinear and fully nonlinear differential equations are relevant classes of such equations and have been widely examined in the mathematical literature. In this work we present a new family of differential equations called "implicit partial differential equations", described in detail in the introduction (c.f. Chapter 1). It is a class of nonlinear equations that does not include the family of fully nonlinear elliptic pdes. We present a new functional analytic method based on the Baire category theorem for handling the existence of almost everywhere solutions of these implicit equations. The results have been obtained for the most part in recent years and have important applications to the calculus of variations, nonlin ear elasticity, problems of phase transitions and optimal design; some results have not been published elsewhere.
Author : Carlo Miranda
Publisher : Springer-Verlag
Page : 385 pages
File Size : 55,8 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783662351475
Author : S. Carl,Seppo Heikkila
Publisher : CRC Press
Page : 336 pages
File Size : 53,7 Mb
Release : 2000-06-14
Category : Mathematics
ISBN : 9781482280951
Extremality results proved in this Monograph for an abstract operator equation provide the theoretical framework for developing new methods that allow the treatment of a variety of discontinuous initial and boundary value problems for both ordinary and partial differential equations, in explicit and implicit forms. By means of these extremality results, the authors prove the existence of extremal solutions between appropriate upper and lower solutions of first and second order discontinuous implicit and explicit ordinary and functional differential equations. They then study the dependence of these extremal solutions on the data. The authors begin by developing an existence theory for an abstract operator equation in ordered spaces and offer new tools for dealing with different kinds of discontinuous implicit and explicit differential equation problems. They present a unified approach to the existence of extremal solutions of quasilinear elliptic and parabolic problems and extend the upper and lower solution method to elliptic and parabolic inclusion of hemivariation type using variational and nonvariational methods. Nonlinear Differential Equations in Ordered Spaces includes research that appears for the first time in book form and is designed as a source book for pure and applied mathematicians. Its self-contained presentation along with numerous worked examples and complete, detailed proofs also make it accessible to researchers in engineering as well as advanced students in these fields.
Author : Jan Malý,William P. Ziemer
Publisher : American Mathematical Soc.
Page : 309 pages
File Size : 50,8 Mb
Release : 1997
Category : Boundary value problems
ISBN : 9780821803356
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 : Carlos E. Kenig
Publisher : American Mathematical Soc.
Page : 146 pages
File Size : 44,9 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821803097
In recent years, there has been a great deal of activity in the study of boundary value problems with minimal smoothness assumptions on the coefficients or on the boundary of the domain in question. These problems are of interest both because of their theoretical importance and the implications for applications, and they have turned out to have profound and fascinating connections with many areas of analysis. Techniques from harmonic analysis have proved to be extremely useful in these studies, both as concrete tools in establishing theorems and as models which suggest what kind of result might be true. Kenig describes these developments and connections for the study of classical boundary value problems on Lipschitz domains and for the corresponding problems for second order elliptic equations in divergence form. He also points out many interesting problems in this area which remain open.
Author : Vicentiu D. Radulescu,Vicenţiu D. Rădulescu
Publisher : Hindawi Publishing Corporation
Page : 205 pages
File Size : 46,7 Mb
Release : 2008
Category : Differential equations, Elliptic
ISBN : 9789774540394
This book provides a comprehensive introduction to the mathematical theory of nonlinear problems described by elliptic partial differential equations. These equations can be seen as nonlinear versions of the classical Laplace equation, and they appear as mathematical models in different branches of physics, chemistry, biology, genetics, and engineering and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on the calculus of variations and functional analysis. Concentrating on single-valued or multivalued elliptic equations with nonlinearities of various types, the aim of this volume is to obtain sharp existence or nonexistence results, as well as decay rates for general classes of solutions. Many technically relevant questions are presented and analyzed in detail. A systematic picture of the most relevant phenomena is obtained for the equations under study, including bifurcation, stability, asymptotic analysis, and optimal regularity of solutions. The method of presentation should appeal to readers with different backgrounds in functional analysis and nonlinear partial differential equations. All chapters include detailed heuristic arguments providing thorough motivation of the study developed later on in the text, in relationship with concrete processes arising in applied sciences. A systematic description of the most relevant singular phenomena described in this volume includes existence (or nonexistence) of solutions, unicity or multiplicity properties, bifurcation and asymptotic analysis, and optimal regularity. The book includes an extensive bibliography and a rich index, thus allowing for quick orientation among the vast collection of literature on the mathematical theory of nonlinear phenomena described by elliptic partial differential equations.
Author : Hervé Le Dret
Publisher : Springer
Page : 253 pages
File Size : 45,8 Mb
Release : 2018-05-25
Category : Mathematics
ISBN : 9783319783901
This textbook presents the essential parts of the modern theory of nonlinear partial differential equations, including the calculus of variations. After a short review of results in real and functional analysis, the author introduces the main mathematical techniques for solving both semilinear and quasilinear elliptic PDEs, and the associated boundary value problems. Key topics include infinite dimensional fixed point methods, the Galerkin method, the maximum principle, elliptic regularity, and the calculus of variations. Aimed at graduate students and researchers, this textbook contains numerous examples and exercises and provides several comments and suggestions for further study.
Author : David Gilbarg,Neil S. Trudinger
Publisher : Springer Science & Business Media
Page : 544 pages
File Size : 49,7 Mb
Release : 2001-01-12
Category : Mathematics
ISBN : 3540411607
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 : Anonim
Publisher : Unknown
Page : 984 pages
File Size : 42,9 Mb
Release : 2008
Category : Mathematics
ISBN : UOM:39015082440606
Author : Veron Laurent
Publisher : World Scientific
Page : 476 pages
File Size : 55,8 Mb
Release : 2017-05-05
Category : Mathematics
ISBN : 9789814730341
This book is devoted to the study of elliptic second-order degenerate quasilinear equations, the model of which is the p-Laplacian, with or without dominant lower order reaction term. Emphasis is put on three aspects: The existence of separable singular solutions enables the description of isolated singularities of general solutions. The construction of singular solutions is delicate and cannot be done without the understanding of the spherical p-harmonic eigenvalue problem.When the equations are considered on a Riemannian manifold, existence or non-existence of solutions depends on geometric assumptions such as the curvature. A priori estimates and Liouville type problems are analyzed.When the equations are considered with a forcing term in the class of measures, their study is strongly linked to the properties of a class of potentials appearing in harmonic analysis such as the Riesz, the Bessel or the Wolff potentials and to their associated capacities. Necessary and sufficient conditions for existence of solutions link the continuity of the measure with respect to some appropriate capacity.
Author : Luis A. Caffarelli,Xavier Cabré
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 52,5 Mb
Release : 1995
Category : Differential equations, Elliptic
ISBN : 9780821804377
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.
Author : J. P. Gossez,Denis Bonheure
Publisher : American Mathematical Soc.
Page : 278 pages
File Size : 50,8 Mb
Release : 2011
Category : Differential equations, Elliptic
ISBN : 9780821849071
This volume contains papers on semi-linear and quasi-linear elliptic equations from the workshop on Nonlinear Elliptic Partial Differential Equations, in honor of Jean-Pierre Gossez's 65th birthday, held September 2-4, 2009 at the Universite Libre de Bruxelles, Belgium. The workshop reflected Gossez's contributions in nonlinear elliptic PDEs and provided an opening to new directions in this very active research area. Presentations covered recent progress in Gossez's favorite topics, namely various problems related to the $p$-Laplacian operator, the antimaximum principle, the Fucik Spectrum, and other related subjects. This volume will be of principle interest to researchers in nonlinear analysis, especially in partial differential equations of elliptic type.