Fully Nonlinear Elliptic Equations

Author : Luis A. Caffarelli,Xavier Cabré
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 54,6 Mb
Release : 1995
Category : Mathematics
ISBN : 9780821804377

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Fully Nonlinear Elliptic Equations by Luis A. Caffarelli,Xavier Cabré Pdf

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 : Qing Han
Publisher : American Mathematical Soc.
Page : 368 pages
File Size : 44,5 Mb
Release : 2016-04-15
Category : Differential equations, Elliptic
ISBN : 9781470426071

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Nonlinear Elliptic Equations of the Second Order by Qing Han Pdf

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.

Author : Nikolai Nadirashvili, Vladimir Tkachev, Serge Vlăduţ
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 46,7 Mb
Release : 2014-12-03
Category : Mathematics
ISBN : 9781470417109

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Nonlinear Elliptic Equations and Nonassociative Algebras by Nikolai Nadirashvili, Vladimir Tkachev, Serge Vlăduţ Pdf

This book presents applications of noncommutative and nonassociative algebras to constructing unusual (nonclassical and singular) solutions to fully nonlinear elliptic partial differential equations of second order. The methods described in the book are used to solve a longstanding problem of the existence of truly weak, nonsmooth viscosity solutions. Moreover, the authors provide an almost complete description of homogeneous solutions to fully nonlinear elliptic equations. It is shown that even in the very restricted setting of "Hessian equations", depending only on the eigenvalues of the Hessian, these equations admit homogeneous solutions of all orders compatible with known regularity for viscosity solutions provided the space dimension is five or larger. To the contrary, in dimension four or less the situation is completely different, and our results suggest strongly that there are no nonclassical homogeneous solutions at all in dimensions three and four. Thus this book gives a complete list of dimensions where nonclassical homogeneous solutions to fully nonlinear uniformly elliptic equations do exist; this should be compared with the situation of, say, ten years ago when the very existence of nonclassical viscosity solutions was not known.

Author : N.V. Krylov
Publisher : Springer
Page : 0 pages
File Size : 52,6 Mb
Release : 2001-11-30
Category : Mathematics
ISBN : 140200334X

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Nonlinear Elliptic and Parabolic Equations of the Second Order by N.V. Krylov Pdf

Approach your problems from the It isn't that they can't see the right end and begin with the solution. It is that they can't see answers. Then one day, perhaps the problem. you will find the final question. G.K. Chesterton. The Scandal of 'The Hermit Clad in Crane Father Brown 'The Point of a Pin'. Feathers' in R. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of mono graphs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theor.etical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

Author : D. Gilbarg,N. S. Trudinger
Publisher : Springer Science & Business Media
Page : 409 pages
File Size : 50,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783642963797

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Elliptic Partial Differential Equations of Second Order by D. Gilbarg,N. S. Trudinger Pdf

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.

Author : N. V. Krylov
Publisher : American Mathematical Soc.
Page : 441 pages
File Size : 52,9 Mb
Release : 2018-09-07
Category : Differential equations, Parabolic
ISBN : 9781470447403

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Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations by N. V. Krylov Pdf

This book concentrates on first boundary-value problems for fully nonlinear second-order uniformly elliptic and parabolic equations with discontinuous coefficients. We look for solutions in Sobolev classes, local or global, or for viscosity solutions. Most of the auxiliary results, such as Aleksandrov's elliptic and parabolic estimates, the Krylov–Safonov and the Evans–Krylov theorems, are taken from old sources, and the main results were obtained in the last few years. Presentation of these results is based on a generalization of the Fefferman–Stein theorem, on Fang-Hua Lin's like estimates, and on the so-called “ersatz” existence theorems, saying that one can slightly modify “any” equation and get a “cut-off” equation that has solutions with bounded derivatives. These theorems allow us to prove the solvability in Sobolev classes for equations that are quite far from the ones which are convex or concave with respect to the Hessians of the unknown functions. In studying viscosity solutions, these theorems also allow us to deal with classical approximating solutions, thus avoiding sometimes heavy constructions from the usual theory of viscosity solutions.

Author : Hervé Le Dret
Publisher : Springer
Page : 253 pages
File Size : 44,9 Mb
Release : 2018-05-25
Category : Mathematics
ISBN : 9783319783901

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Nonlinear Elliptic Partial Differential Equations by Hervé Le Dret Pdf

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 : Wenxiong Chen,Congming Li
Publisher : Unknown
Page : 0 pages
File Size : 50,8 Mb
Release : 2010
Category : Differential equations, Elliptic
ISBN : 1601330065

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Methods on Nonlinear Elliptic Equations by Wenxiong Chen,Congming Li Pdf

Author : Michel Chipot
Publisher : Springer Science & Business Media
Page : 289 pages
File Size : 44,9 Mb
Release : 2009-02-19
Category : Mathematics
ISBN : 9783764399818

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Elliptic Equations: An Introductory Course by Michel Chipot Pdf

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.

Author : Ya-Zhe Chen,Lan-Cheng Wu
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 55,8 Mb
Release : 1998
Category : Mathematics
ISBN : 9780821819241

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Second Order Elliptic Equations and Elliptic Systems by Ya-Zhe Chen,Lan-Cheng Wu Pdf

There are two parts to the book. In the first part, a complete introduction of various kinds of a priori estimate methods for the Dirichlet problem of second order elliptic partial differential equations is presented. In the second part, the existence and regularity theories of the Dirichlet problem for linear and nonlinear second order elliptic partial differential systems are introduced. The book features appropriate materials and is an excellent textbook for graduate students. The volume is also useful as a reference source for undergraduate mathematics majors, graduate students, professors, and scientists.

Author : Sun-Yung A. Chang
Publisher : European Mathematical Society
Page : 106 pages
File Size : 42,7 Mb
Release : 2004
Category : Computers
ISBN : 303719006X

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Non-linear Elliptic Equations in Conformal Geometry by Sun-Yung A. Chang Pdf

Non-linear elliptic partial differential equations are an important tool in the study of Riemannian metrics in differential geometry, in particular for problems concerning the conformal change of metrics in Riemannian geometry. In recent years the role played by the second order semi-linear elliptic equations in the study of Gaussian curvature and scalar curvature has been extended to a family of fully non-linear elliptic equations associated with other symmetric functions of the Ricci tensor. A case of particular interest is the second symmetric function of the Ricci tensor in dimension four closely related to the Pfaffian. In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g., higher order conformal invariant operators, Sobolev inequalities, blow-up analysis) in order to solve a fully nonlinear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four. The material is suitable for graduate students and research mathematicians interested in geometry, topology, and differential equations.

Author : Jingyi Chen,Peng Lu,Zhiqin Lu,Zhou Zhang
Publisher : Springer Nature
Page : 616 pages
File Size : 40,5 Mb
Release : 2020-04-10
Category : Mathematics
ISBN : 9783030349530

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Geometric Analysis by Jingyi Chen,Peng Lu,Zhiqin Lu,Zhou Zhang Pdf

This edited volume has a two-fold purpose. First, comprehensive survey articles provide a way for beginners to ease into the corresponding sub-fields. These are then supplemented by original works that give the more advanced readers a glimpse of the current research in geometric analysis and related PDEs. The book is of significant interest for researchers, including advanced Ph.D. students, working in geometric analysis. Readers who have a secondary interest in geometric analysis will benefit from the survey articles. The results included in this book will stimulate further advances in the subjects: geometric analysis, including complex differential geometry, symplectic geometry, PDEs with a geometric origin, and geometry related to topology. Contributions by Claudio Arezzo, Alberto Della Vedova, Werner Ballmann, Henrik Matthiesen, Panagiotis Polymerakis, Sun-Yung A. Chang, Zheng-Chao Han, Paul Yang, Tobias Holck Colding, William P. Minicozzi II, Panagiotis Dimakis, Richard Melrose, Akito Futaki, Hajime Ono, Jiyuan Han, Jeff A. Viaclovsky, Bruce Kleiner, John Lott, Sławomir Kołodziej, Ngoc Cuong Nguyen, Chi Li, Yuchen Liu, Chenyang Xu, YanYan Li, Luc Nguyen, Bo Wang, Shiguang Ma, Jie Qing, Xiaonan Ma, Sean Timothy Paul, Kyriakos Sergiou, Tristan Rivière, Yanir A. Rubinstein, Natasa Sesum, Jian Song, Jeffrey Streets, Neil S. Trudinger, Yu Yuan, Weiping Zhang, Xiaohua Zhu and Aleksey Zinger.

Author : Arina A. Arkhipova,Alexander I. Nazarov,Nina Nikolaevna Uralʹt︠s︡eva
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 52,5 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821849972

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Nonlinear Partial Differential Equations and Related Topics by Arina A. Arkhipova,Alexander I. Nazarov,Nina Nikolaevna Uralʹt︠s︡eva Pdf

"St. Petersburg PDE seminar, special session dedicated to N.N. Uraltseva's [75th] anniversary, June 2009"--P. [vi].

Author : Patrick Fitzpatrick,Jacobo Pejsachowicz
Publisher : American Mathematical Soc.
Page : 145 pages
File Size : 50,8 Mb
Release : 1993-01-01
Category : Mathematics
ISBN : 9780821825440

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Orientation and the Leray-Schauder Theory for Fully Nonlinear Elliptic Boundary Value Problems by Patrick Fitzpatrick,Jacobo Pejsachowicz Pdf

The aim of this work is to develop an additive, integer-valued degree theory for the class of quasilinear Fredholm mappings. This class is sufficiently large that, within its framework, one can study general fully nonlinear elliptic boundary value problems. A degree for the whole class of quasilinear Fredholm mappings must necessarily accommodate sign-switching of the degree along admissible homotopies. The authors introduce ''parity'', a homotopy invariant of paths of linear Fredholm operators having invertible endpoints. The parity provides a complete description of the possible changes in sign of the degree and thereby permits use of the degree to prove multiplicity and bifurcation theorems for quasilinear Fredholm mappings. Applications are given to the study of fully nonlinear elliptic boundary value problems.

Author : Nikolaĭ Vladimirovich Krylov
Publisher : American Mathematical Soc.
Page : 164 pages
File Size : 46,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821805695

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Lectures on Elliptic and Parabolic Equations in Hölder Spaces by Nikolaĭ Vladimirovich Krylov Pdf

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.