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Nonlinear Elliptic Partial Differential Equations
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Book Synopsis Nonlinear Elliptic Partial Differential Equations by : Hervé Le Dret
Download or read book Nonlinear Elliptic Partial Differential Equations written by Hervé Le Dret and published by Springer. This book was released on 2018-05-25 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Nonlinear Elliptic Equations of the Second Order by : Qing Han
Download or read book Nonlinear Elliptic Equations of the Second Order written by Qing Han and published by American Mathematical Soc.. This book was released on 2016-04-15 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear elliptic differential equations are a diverse subject with important applications to the physical and social sciences and engineering. They also arise naturally in geometry. In particular, much of the progress in the area in the twentieth century was driven by geometric applications, from the Bernstein problem to the existence of Kähler–Einstein metrics. This book, designed as a textbook, provides a detailed discussion of the Dirichlet problems for quasilinear and fully nonlinear elliptic differential equations of the second order with an emphasis on mean curvature equations and on Monge–Ampère equations. It gives a user-friendly introduction to the theory of nonlinear elliptic equations with special attention given to basic results and the most important techniques. Rather than presenting the topics in their full generality, the book aims at providing self-contained, clear, and “elementary” proofs for results in important special cases. This book will serve as a valuable resource for graduate students or anyone interested in this subject.
Book Synopsis Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations by : Vicentiu D. Radulescu
Download or read book Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations written by Vicentiu D. Radulescu and published by Hindawi Publishing Corporation. This book was released on 2008 with total page 205 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Elliptic Partial Differential Equations by : Lucio Boccardo
Download or read book Elliptic Partial Differential Equations written by Lucio Boccardo and published by Walter de Gruyter. This book was released on 2013-10-29 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Elliptic Partial Differential Equations by : Qing Han
Download or read book Elliptic Partial Differential Equations written by Qing Han and published by American Mathematical Soc.. This book was released on 2011 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Methods on Nonlinear Elliptic Equations by : Wenxiong Chen
Download or read book Methods on Nonlinear Elliptic Equations written by Wenxiong Chen and published by . This book was released on 2010 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Fully Nonlinear Elliptic Equations by : Luis A. Caffarelli
Download or read book Fully Nonlinear Elliptic Equations written by Luis A. Caffarelli and published by American Mathematical Soc.. This book was released on 1995 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Introduction to Nonlinear Partial Differential Equations by : J. David Logan
Download or read book An Introduction to Nonlinear Partial Differential Equations written by J. David Logan and published by John Wiley & Sons. This book was released on 2008-04-11 with total page 416 pages. Available in PDF, EPUB and Kindle. Book excerpt: Praise for the First Edition: "This book is well conceived and well written. The author has succeeded in producing a text on nonlinear PDEs that is not only quite readable but also accessible to students from diverse backgrounds." —SIAM Review A practical introduction to nonlinear PDEs and their real-world applications Now in a Second Edition, this popular book on nonlinear partial differential equations (PDEs) contains expanded coverage on the central topics of applied mathematics in an elementary, highly readable format and is accessible to students and researchers in the field of pure and applied mathematics. This book provides a new focus on the increasing use of mathematical applications in the life sciences, while also addressing key topics such as linear PDEs, first-order nonlinear PDEs, classical and weak solutions, shocks, hyperbolic systems, nonlinear diffusion, and elliptic equations. Unlike comparable books that typically only use formal proofs and theory to demonstrate results, An Introduction to Nonlinear Partial Differential Equations, Second Edition takes a more practical approach to nonlinear PDEs by emphasizing how the results are used, why they are important, and how they are applied to real problems. The intertwining relationship between mathematics and physical phenomena is discovered using detailed examples of applications across various areas such as biology, combustion, traffic flow, heat transfer, fluid mechanics, quantum mechanics, and the chemical reactor theory. New features of the Second Edition also include: Additional intermediate-level exercises that facilitate the development of advanced problem-solving skills New applications in the biological sciences, including age-structure, pattern formation, and the propagation of diseases An expanded bibliography that facilitates further investigation into specialized topics With individual, self-contained chapters and a broad scope of coverage that offers instructors the flexibility to design courses to meet specific objectives, An Introduction to Nonlinear Partial Differential Equations, Second Edition is an ideal text for applied mathematics courses at the upper-undergraduate and graduate levels. It also serves as a valuable resource for researchers and professionals in the fields of mathematics, biology, engineering, and physics who would like to further their knowledge of PDEs.
Book Synopsis Lectures on Elliptic Partial Differential Equations by : Luigi Ambrosio
Download or read book Lectures on Elliptic Partial Differential Equations written by Luigi Ambrosio and published by Springer. This book was released on 2019-01-10 with total page 230 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book originates from the Elliptic PDE course given by the first author at the Scuola Normale Superiore in recent years. It covers the most classical aspects of the theory of Elliptic Partial Differential Equations and Calculus of Variations, including also more recent developments on partial regularity for systems and the theory of viscosity solutions.
Book Synopsis Nonlinear Partial Differential Equations with Applications by : Tomás Roubicek
Download or read book Nonlinear Partial Differential Equations with Applications written by Tomás Roubicek and published by Springer Science & Business Media. This book was released on 2006-01-17 with total page 415 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book primarily concerns quasilinear and semilinear elliptic and parabolic partial differential equations, inequalities, and systems. The exposition quickly leads general theory to analysis of concrete equations, which have specific applications in such areas as electrically (semi-) conductive media, modeling of biological systems, and mechanical engineering. Methods of Galerkin or of Rothe are exposed in a large generality.
Book Synopsis Boundary Value Problems for Nonlinear Elliptic Equations in Divergence Form by : Abubakar Mwasa
Download or read book Boundary Value Problems for Nonlinear Elliptic Equations in Divergence Form written by Abubakar Mwasa and published by Linköping University Electronic Press. This book was released on 2021-02-23 with total page 22 pages. Available in PDF, EPUB and Kindle. Book excerpt: The thesis consists of three papers focussing on the study of nonlinear elliptic partial differential equations in a nonempty open subset Ω of the n-dimensional Euclidean space Rn. We study the existence and uniqueness of the solutions, as well as their behaviour near the boundary of Ω. The behaviour of the solutions at infinity is also discussed when Ω is unbounded. In Paper A, we consider a mixed boundary value problem for the p-Laplace equation ∆pu := div(|∇u| p−2∇u) = 0 in an open infinite circular half-cylinder with prescribed Dirichlet boundary data on a part of the boundary and zero Neumann boundary data on the rest. By a suitable transformation of the independent variables, this mixed problem is transformed into a Dirichlet problem for a degenerate (weighted) elliptic equation on a bounded set. By analysing the transformed problem in weighted Sobolev spaces, it is possible to obtain the existence of continuous weak solutions to the mixed problem, both for Sobolev and for continuous data on the Dirichlet part of the boundary. A characterisation of the boundary regularity of the point at infinity is obtained in terms of a new variational capacity adapted to the cylinder. In Paper B, we study Perron solutions to the Dirichlet problem for the degenerate quasilinear elliptic equation div A(x, ∇u) = 0 in a bounded open subset of Rn. The vector-valued function A satisfies the standard ellipticity assumptions with a parameter 1 < p < ∞ and a p-admissible weight w. For general boundary data, the Perron method produces a lower and an upper solution, and if they coincide then the boundary data are called resolutive. We show that arbitrary perturbations on sets of weighted p-capacity zero of continuous (and quasicontinuous Sobolev) boundary data f are resolutive, and that the Perron solutions for f and such perturbations coincide. As a consequence, it is also proved that the Perron solution with continuous boundary data is the unique bounded continuous weak solution that takes the required boundary data outside a set of weighted p-capacity zero. Some results in Paper C are a generalisation of those in Paper A, extended to quasilinear elliptic equations of the form div A(x, ∇u) = 0. Here, results from Paper B are used to prove the existence and uniqueness of continuous weak solutions to the mixed boundary value problem for continuous Dirichlet data. Regularity of the boundary point at infinity for the equation div A(x, ∇u) = 0 is characterised by a Wiener type criterion. We show that sets of Sobolev p-capacity zero are removable for the solutions and also discuss the behaviour of the solutions at ∞. In particular, a certain trichotomy is proved, similar to the Phragmén–Lindelöf principle.
Book Synopsis Contributions to Nonlinear Elliptic Equations and Systems by : Alexandre N. Carvalho
Download or read book Contributions to Nonlinear Elliptic Equations and Systems written by Alexandre N. Carvalho and published by Birkhäuser. This book was released on 2015-11-14 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume of contributions pays tribute to the life and work of Djairo Guedes de Figueiredo on the occasion of his 80th birthday. The articles it contains were born out of the ICMC Summer Meeting on Differential Equations – 2014 Chapter, also dedicated to de Figueiredo and held at the Universidade de São Paulo at São Carlos, Brazil from February 3-7, 2014. The contributing authors represent a group of international experts in the field and discuss recent trends and new directions in nonlinear elliptic partial differential equations and systems. Djairo Guedes de Figueiredo has had a very active scientific career, publishing 29 monographs and over one hundred research articles. His influence on Brazilian mathematics has made him one of the pillars of the subject in that country. He had a major impact on the development of analysis, especially in its application to nonlinear elliptic partial differential equations and systems throughout the entire world. The articles collected here pay tribute to him and his legacy and are intended for graduate students and researchers in mathematics and related areas who are interested in nonlinear elliptic partial differential equations and systems.
Book Synopsis Elliptic Partial Differential Equations by : Vitaly Volpert
Download or read book Elliptic Partial Differential Equations written by Vitaly Volpert and published by Springer Science & Business Media. This book was released on 2011-03-03 with total page 649 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic partial differential equations has undergone an important development over the last two centuries. Together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications, it has become one of the most richly enhanced fields of mathematics. This monograph undertakes a systematic presentation of the theory of general elliptic operators. The author discusses a priori estimates, normal solvability, the Fredholm property, the index of an elliptic operator, operators with a parameter, and nonlinear Fredholm operators. Particular attention is paid to elliptic problems in unbounded domains which have not yet been sufficiently treated in the literature and which require some special approaches. The book also contains an analysis of non-Fredholm operators and discrete operators as well as extensive historical and bibliographical comments . The selected topics and the author's level of discourse will make this book a most useful resource for researchers and graduate students working in the broad field of partial differential equations and applications.
Book Synopsis Elliptic Partial Differential Equations of Second Order by : D. Gilbarg
Download or read book Elliptic Partial Differential Equations of Second Order written by D. Gilbarg and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is intended as an essentially self contained exposition of portions of the theory of second order quasilinear elliptic partial differential equations, with emphasis on the Dirichlet problem in bounded domains. It grew out of lecture notes for graduate courses by the authors at Stanford University, the final material extending well beyond the scope of these courses. By including preparatory chapters on topics such as potential theory and functional analysis, we have attempted to make the work accessible to a broad spectrum of readers. Above all, we hope the readers of this book will gain an appreciation of the multitude of ingenious barehanded techniques that have been developed in the study of elliptic equations and have become part of the repertoire of analysis. Many individuals have assisted us during the evolution of this work over the past several years. In particular, we are grateful for the valuable discussions with L. M. Simon and his contributions in Sections 15.4 to 15.8; for the helpful comments and corrections of J. M. Cross, A. S. Geue, J. Nash, P. Trudinger and B. Turkington; for the contributions of G. Williams in Section 10.5 and of A. S. Geue in Section 10.6; and for the impeccably typed manuscript which resulted from the dedicated efforts oflsolde Field at Stanford and Anna Zalucki at Canberra. The research of the authors connected with this volume was supported in part by the National Science Foundation.
Book Synopsis Fine Regularity of Solutions of Elliptic Partial Differential Equations by : Jan Malý
Download or read book Fine Regularity of Solutions of Elliptic Partial Differential Equations written by Jan Malý and published by American Mathematical Soc.. This book was released on 1997 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Nonlinear Elliptic Partial Differential Equations by : J. P. Gossez
Download or read book Nonlinear Elliptic Partial Differential Equations written by J. P. Gossez and published by American Mathematical Soc.. This book was released on 2011 with total page 278 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Regularity Results for Nonlinear Elliptic Systems and Applications by : Alain Bensoussan
Download or read book Regularity Results for Nonlinear Elliptic Systems and Applications written by Alain Bensoussan and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 450 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects many helpful techniques for obtaining regularity results for solutions of nonlinear systems of partial differential equations. These are applied in various cases to provide useful examples and relevant results, particularly in such fields as fluid mechanics, solid mechanics, semiconductor theory and game theory.
