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Author : Edgard A. Pimentel
Publisher : Cambridge University Press
Page : 203 pages
File Size : 47,9 Mb
Release : 2022-09-29
Category : Mathematics
ISBN : 9781009096669
A modern account of elliptic regularity theory, with a rigorous presentation of recent developments for fundamental models.
Author : Edgard A. Pimentel
Publisher : Cambridge University Press
Page : 204 pages
File Size : 42,9 Mb
Release : 2022-06-30
Category : Mathematics
ISBN : 9781009103121
Presenting the basics of elliptic PDEs in connection with regularity theory, the book bridges fundamental breakthroughs – such as the Krylov–Safonov and Evans–Krylov results, Caffarelli's regularity theory, and the counterexamples due to Nadirashvili and Vlăduţ – and modern developments, including improved regularity for flat solutions and the partial regularity result. After presenting this general panorama, accounting for the subtleties surrounding C-viscosity and Lp-viscosity solutions, the book examines important models through approximation methods. The analysis continues with the asymptotic approach, based on the recession operator. After that, approximation techniques produce a regularity theory for the Isaacs equation, in Sobolev and Hölder spaces. Although the Isaacs operator lacks convexity, approximation methods are capable of producing Hölder continuity for the Hessian of the solutions by connecting the problem with a Bellman equation. To complete the book, degenerate models are studied and their optimal regularity is described.
Author : Lisa Beck
Publisher : Springer
Page : 201 pages
File Size : 40,6 Mb
Release : 2016-04-08
Category : Mathematics
ISBN : 9783319274850
These lecture notes provide a self-contained introduction to regularity theory for elliptic equations and systems in divergence form. After a short review of some classical results on everywhere regularity for scalar-valued weak solutions, the presentation focuses on vector-valued weak solutions to a system of several coupled equations. In the vectorial case, weak solutions may have discontinuities and so are expected, in general, to be regular only outside of a set of measure zero. Several methods are presented concerning the proof of such partial regularity results, and optimal regularity is discussed. Finally, a short overview is given on the current state of the art concerning the size of the singular set on which discontinuities may occur. The notes are intended for graduate and postgraduate students with a solid background in functional analysis and some familiarity with partial differential equations; they will also be of interest to researchers working on related topics.
Author : John A. Trangenstein
Publisher : Cambridge University Press
Page : 657 pages
File Size : 40,6 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9780521877268
For mathematicians and engineers interested in applying numerical methods to physical problems this book is ideal. Numerical ideas are connected to accompanying software, which is also available online. By seeing the complete description of the methods in both theory and implementation, students will more easily gain the knowledge needed to write their own application programs or develop new theory. The book contains careful development of the mathematical tools needed for analysis of the numerical methods, including elliptic regularity theory and approximation theory. Variational crimes, due to quadrature, coordinate mappings, domain approximation and boundary conditions, are analyzed. The claims are stated with full statement of the assumptions and conclusions, and use subscripted constants which can be traced back to the origination (particularly in the electronic version, which can be found on the accompanying CD-ROM).
Author : Shuhong Chen,Zhong Tan
Publisher : LAP Lambert Academic Publishing
Page : 296 pages
File Size : 54,7 Mb
Release : 2012
Category : Analytic functions
ISBN : 3659278068
Regularity theory is one of the most challenging problems in modern theory of partial differential equations. It has attracted peoples' eyes for a long history. A classical method of partial regularity theory is the "freezing the coefficients" method. The proof is complex and troublesome. And the result obtained by this method is not optimal. In this book, we use the method of A-harmonic approximation, to consider regularity theory for nonlinear partial differential systems. The new method not only allows one to simplify the procedure of proof, but also to establish optimal regularity results directly. This book should be useful to professionals in partial differential equations.
Author : Hervé Le Dret,Brigitte Lucquin
Publisher : Birkhäuser
Page : 395 pages
File Size : 45,7 Mb
Release : 2016-02-11
Category : Mathematics
ISBN : 9783319270678
This book is devoted to the study of partial differential equation problems both from the theoretical and numerical points of view. After presenting modeling aspects, it develops the theoretical analysis of partial differential equation problems for the three main classes of partial differential equations: elliptic, parabolic and hyperbolic. Several numerical approximation methods adapted to each of these examples are analyzed: finite difference, finite element and finite volumes methods, and they are illustrated using numerical simulation results. Although parts of the book are accessible to Bachelor students in mathematics or engineering, it is primarily aimed at Masters students in applied mathematics or computational engineering. The emphasis is on mathematical detail and rigor for the analysis of both continuous and discrete problems.
Author : Chris Godsil,Hanmeng Zhan
Publisher : Cambridge University Press
Page : 151 pages
File Size : 53,8 Mb
Release : 2022-12-31
Category : Computers
ISBN : 9781009261685
Explore the mathematics arising from discrete quantum walks in this introduction to a rapidly developing area.
Author : Leonid Berlyand,Alexander G. Kolpakov,Alexei Novikov
Publisher : Cambridge University Press
Page : 259 pages
File Size : 49,9 Mb
Release : 2012-12-13
Category : Mathematics
ISBN : 9781139851886
In recent years the traditional subject of continuum mechanics has grown rapidly and many new techniques have emerged. This text provides a rigorous, yet accessible introduction to the basic concepts of the network approximation method and provides a unified approach for solving a wide variety of applied problems. As a unifying theme, the authors discuss in detail the transport problem in a system of bodies. They solve the problem of closely placed bodies using the new method of network approximation for PDE with discontinuous coefficients, developed in the 2000s by applied mathematicians in the USA and Russia. Intended for graduate students in applied mathematics and related fields such as physics, chemistry and engineering, the book is also a useful overview of the topic for researchers in these areas.
Author : John Arthur Trangenstein
Publisher : Unknown
Page : 0 pages
File Size : 43,5 Mb
Release : 2013
Category : Mathematics
ISBN : 1107688078
"For mathematicians and engineers interested in applying numerical methods to physical problems this book is ideal. Numerical ideas are connected to accompanying software, which is also available online. By seeing the complete description of the methods in both theory and implementation, students will more easily gain the knowledge needed to write their own application programs or develop new theory. The book contains careful development of the mathematical tools needed for analysis of the numerical methods, including elliptic regularity theory and approximation theory. Variational crimes, due to quadrature, coordinate mappings, domain approximation and boundary conditions, are analyzed. The claims are stated with full statement of the assumptions and conclusions, and use subscripted constants which can be traced back to the origination (particularly in the electronic version, which is available online and on the accompanying CD-ROM)"--
Author : Klaus Böhmer
Publisher : Oxford University Press
Page : 775 pages
File Size : 54,9 Mb
Release : 2010-10-07
Category : Computers
ISBN : 9780199577040
Boehmer systmatically handles the different numerical methods for nonlinear elliptic problems.
Author : Alexandre Ern,Jean-Luc Guermond
Publisher : Springer Nature
Page : 491 pages
File Size : 48,6 Mb
Release : 2021-04-22
Category : Mathematics
ISBN : 9783030569235
This book is the second volume of a three-part textbook suitable for graduate coursework, professional engineering and academic research. It is also appropriate for graduate flipped classes. Each volume is divided into short chapters. Each chapter can be covered in one teaching unit and includes exercises as well as solutions available from a dedicated website. The salient ideas can be addressed during lecture, with the rest of the content assigned as reading material. To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy. Volume II is divided into 32 chapters plus one appendix. The first part of the volume focuses on the approximation of elliptic and mixed PDEs, beginning with fundamental results on well-posed weak formulations and their approximation by the Galerkin method. The material covered includes key results such as the BNB theorem based on inf-sup conditions, Céa's and Strang's lemmas, and the duality argument by Aubin and Nitsche. Important implementation aspects regarding quadratures, linear algebra, and assembling are also covered. The remainder of Volume II focuses on PDEs where a coercivity property is available. It investigates conforming and nonconforming approximation techniques (Galerkin, boundary penalty, Crouzeix—Raviart, discontinuous Galerkin, hybrid high-order methods). These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows). Finally, the appendix addresses fundamental results on the surjectivity, bijectivity, and coercivity of linear operators in Banach spaces.
Author : Vladimir Dotsenko,Sergey Shadrin,Bruno Vallette
Publisher : Cambridge University Press
Page : 188 pages
File Size : 41,9 Mb
Release : 2023-08-31
Category : Mathematics
ISBN : 9781108967020
Covering an exceptional range of topics, this text provides a unique overview of the Maurer—Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.
Author : Jerome
Publisher : Academic Press
Page : 279 pages
File Size : 54,9 Mb
Release : 1983-04-22
Category : Computers
ISBN : 9780080956701
Approximation of Nonlinear Evolution Systems
Author : Pierre Cardaliaguet,Alessio Porretta,Francesco Salvarani
Publisher : Springer
Page : 218 pages
File Size : 53,6 Mb
Release : 2018-12-22
Category : Mathematics
ISBN : 9783030019471
This volume covers selected topics addressed and discussed during the workshop “PDE models for multi-agent phenomena,” which was held in Rome, Italy, from November 28th to December 2nd, 2016. The content mainly focuses on kinetic equations and mean field games, which provide a solid framework for the description of multi-agent phenomena. The book includes original contributions on the theoretical and numerical study of the MFG system: the uniqueness issue and finite difference methods for the MFG system, MFG with state constraints, and application of MFG to market competition. The book also presents new contributions on the analysis and numerical approximation of the Fokker-Planck-Kolmogorov equations, the isotropic Landau model, the dynamical approach to the quantization problem and the asymptotic methods for fully nonlinear elliptic equations. Chiefly intended for researchers interested in the mathematical modeling of collective phenomena, the book provides an essential overview of recent advances in the field and outlines future research directions.
Author : H. F. Weinberger
Publisher : SIAM
Page : 165 pages
File Size : 55,8 Mb
Release : 1974-01-01
Category : Mathematics
ISBN : 1611970539
Provides a common setting for various methods of bounding the eigenvalues of a self-adjoint linear operator and emphasizes their relationships. A mapping principle is presented to connect many of the methods. The eigenvalue problems studied are linear, and linearization is shown to give important information about nonlinear problems. Linear vector spaces and their properties are used to uniformly describe the eigenvalue problems presented that involve matrices, ordinary or partial differential operators, and integro-differential operators.