Elliptic Regularity Theory By Approximation Methods

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Elliptic Regularity Theory By Approximation Methods

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Author : Edgard A. Pimentel
language : en
Publisher: Cambridge University Press
Release Date : 2022-09-29

Elliptic Regularity Theory By Approximation Methods written by Edgard A. Pimentel and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-09-29 with Mathematics categories.

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.

Direct Methods In The Theory Of Elliptic Equations

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Author : Jindrich Necas
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-10-06

Direct Methods In The Theory Of Elliptic Equations written by Jindrich Necas and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-10-06 with Mathematics categories.

Nečas’ book Direct Methods in the Theory of Elliptic Equations, published 1967 in French, has become a standard reference for the mathematical theory of linear elliptic equations and systems. This English edition, translated by G. Tronel and A. Kufner, presents Nečas’ work essentially in the form it was published in 1967. It gives a timeless and in some sense definitive treatment of a number issues in variational methods for elliptic systems and higher order equations. The text is recommended to graduate students of partial differential equations, postdoctoral associates in Analysis, and scientists working with linear elliptic systems. In fact, any researcher using the theory of elliptic systems will benefit from having the book in his library. The volume gives a self-contained presentation of the elliptic theory based on the "direct method", also known as the variational method. Due to its universality and close connections to numerical approximations, the variational method has become one of the most important approaches to the elliptic theory. The method does not rely on the maximum principle or other special properties of the scalar second order elliptic equations, and it is ideally suited for handling systems of equations of arbitrary order. The prototypical examples of equations covered by the theory are, in addition to the standard Laplace equation, Lame’s system of linear elasticity and the biharmonic equation (both with variable coefficients, of course). General ellipticity conditions are discussed and most of the natural boundary condition is covered. The necessary foundations of the function space theory are explained along the way, in an arguably optimal manner. The standard boundary regularity requirement on the domains is the Lipschitz continuity of the boundary, which "when going beyond the scalar equations of second order" turns out to be a very natural class. These choices reflect the author's opinion that the Lamesystem and the biharmonic equations are just as important as the Laplace equation, and that the class of the domains with the Lipschitz continuous boundary (as opposed to smooth domains) is the most natural class of domains to consider in connection with these equations and their applications.

Numerical Solution Of Elliptic And Parabolic Partial Differential Equations With Cd Rom

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Author : John A. Trangenstein
language : en
Publisher: Cambridge University Press
Release Date : 2013-04-18

Numerical Solution Of Elliptic And Parabolic Partial Differential Equations With Cd Rom written by John A. Trangenstein and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-18 with Mathematics categories.

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).

Introduction To The Network Approximation Method For Materials Modeling

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Author : Leonid Berlyand
language : en
Publisher: Cambridge University Press
Release Date : 2012-12-13

Introduction To The Network Approximation Method For Materials Modeling written by Leonid Berlyand and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-13 with Mathematics categories.

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.

Maurer Cartan Methods In Deformation Theory

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Author : Vladimir Dotsenko
language : en
Publisher: Cambridge University Press
Release Date : 2023-09-07

Maurer Cartan Methods In Deformation Theory written by Vladimir Dotsenko and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-07 with Mathematics categories.

A unique overview of the Maurer-Cartan methods in algebra, geometry, topology, and mathematical physics.

Approximation Of Nonlinear Evolution Systems

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Author : Jerome
language : en
Publisher: Academic Press
Release Date : 1983-04-22

Approximation Of Nonlinear Evolution Systems written by Jerome and has been published by Academic Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1983-04-22 with Computers categories.

Approximation of Nonlinear Evolution Systems

Partial Differential Equations Modeling Analysis And Numerical Approximation

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Author : Hervé Le Dret
language : en
Publisher: Birkhäuser
Release Date : 2016-02-11

Partial Differential Equations Modeling Analysis And Numerical Approximation written by Hervé Le Dret and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-02-11 with Mathematics categories.

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.

A3n2m Approximation Applications And Analysis Of Nonlocal Nonlinear Models

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Author : Tadele Mengesha
language : en
Publisher: Springer Nature
Release Date : 2023-09-12

A3n2m Approximation Applications And Analysis Of Nonlocal Nonlinear Models written by Tadele Mengesha and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-12 with Mathematics categories.

This volume collects papers based on plenary and invited talks given at the 50th Barrett Memorial Lectures on Approximation, Applications, and Analysis of Nonlocal, Nonlinear Models that was organized by the University of Tennessee, Knoxville and held virtually in May 2021. The three-day meeting brought together experts from the computational, scientific, engineering, and mathematical communities who work with nonlocal models. These proceedings collect contributions and give a survey of the state of the art in computational practices, mathematical analysis, applications of nonlocal models, and explorations of new application domains. The volume benefits from the mixture of contributions by computational scientists, mathematicians, and application specialists. The content is suitable for graduate students as well as specialists working with nonlocal models and covers topics on fractional PDEs, regularity theory for kinetic equations, approximation theory for fractional diffusion, analysis of nonlocal diffusion model as a bridge between local and fractional PDEs, and more.

Splines And Pdes From Approximation Theory To Numerical Linear Algebra

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Author : Angela Kunoth
language : en
Publisher: Springer
Release Date : 2018-09-20

Splines And Pdes From Approximation Theory To Numerical Linear Algebra written by Angela Kunoth and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-20 with Mathematics categories.

This book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.

The Mathematical Theory Of Finite Element Methods

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Author : Susanne Brenner
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-17

The Mathematical Theory Of Finite Element Methods written by Susanne Brenner and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-17 with Mathematics categories.

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math (AMS) series, which will focus on advanced textbooks ematical Sciences and research level monographs. Preface This book develops the basic mathematical theory of the finite element method, the most widely used technique for engineering design and analysis. One purpose of this book is to formalize basic tools that are commonly used by researchers in the field but never published. It is intended primarily for mathematics graduate students and mathematically sophisticated engineers and scientists. The book has been the basis for graduate-level courses at The Uni versity of Michigan, Penn State University and the University of Houston.