Lectures on Elliptic Boundary Value Problems

Lectures on Elliptic Boundary Value Problems
Author: Shmuel Agmon
Publisher: American Mathematical Soc.
Total Pages: 225
Release: 2010-02-03
Genre: Mathematics
ISBN: 0821849107

This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality.

Lectures on Elliptic and Parabolic Equations in Holder Spaces

Lectures on Elliptic and Parabolic Equations in Holder Spaces
Author: Nikolaĭ Vladimirovich Krylov
Publisher: American Mathematical Soc.
Total Pages: 178
Release: 1996
Genre: Mathematics
ISBN: 082180569X

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.

Lectures on Elliptic and Parabolic Equations in Sobolev Spaces

Lectures on Elliptic and Parabolic Equations in Sobolev Spaces
Author: Nikolaĭ Vladimirovich Krylov
Publisher: American Mathematical Soc.
Total Pages: 377
Release: 2008
Genre: Mathematics
ISBN: 0821846841

This book concentrates on the basic facts and ideas of the modern theory of linear elliptic and parabolic equations in Sobolev spaces. The main areas covered in this book are the first boundary-value problem for elliptic equations and the Cauchy problem for parabolic equations. In addition, other boundary-value problems such as the Neumann or oblique derivative problems are briefly covered. As is natural for a textbook, the main emphasis is on organizing well-known ideas in a self-contained exposition. Among the topics included that are not usually covered in a textbook are a relatively recent development concerning equations with $\textsf{VMO}$ coefficients and the study of parabolic equations with coefficients measurable only with respect to the time variable. There are numerous exercises which help the reader better understand the material. After going through the book, the reader will have a good understanding of results available in the modern theory of partial differential equations and the technique used to obtain them. Prerequesites are basics of measure theory, the theory of $L p$ spaces, and the Fourier transform.

Lectures on Linear Partial Differential Equations

Lectures on Linear Partial Differential Equations
Author: Grigoriĭ Ilʹich Eskin
Publisher: American Mathematical Soc.
Total Pages: 432
Release: 2011
Genre: Mathematics
ISBN: 0821852841

This is a reader-friendly, relatively short introduction to the modern theory of linear partial differential equations. An effort has been made to present complete proofs in an accessible and self-contained form. The first three chapters are on elementary distribution theory and Sobolev spaces. The following chapters study the Cauchy problem for parabolic and hyperbolic equations, boundary value problems for elliptic equations, heat trace asymptotics, and scattering theory.

Hierarchical Matrices

Hierarchical Matrices
Author: Mario Bebendorf
Publisher: Springer Science & Business Media
Total Pages: 303
Release: 2008-06-25
Genre: Mathematics
ISBN: 3540771476

Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.

Lectures on Differential Equations

Lectures on Differential Equations
Author: Philip L. Korman
Publisher: American Mathematical Soc.
Total Pages: 414
Release: 2019-08-30
Genre: Mathematics
ISBN: 1470451735

Lectures on Differential Equations provides a clear and concise presentation of differential equations for undergraduates and beginning graduate students. There is more than enough material here for a year-long course. In fact, the text developed from the author's notes for three courses: the undergraduate introduction to ordinary differential equations, the undergraduate course in Fourier analysis and partial differential equations, and a first graduate course in differential equations. The first four chapters cover the classical syllabus for the undergraduate ODE course leavened by a modern awareness of computing and qualitative methods. The next two chapters contain a well-developed exposition of linear and nonlinear systems with a similarly fresh approach. The final two chapters cover boundary value problems, Fourier analysis, and the elementary theory of PDEs. The author makes a concerted effort to use plain language and to always start from a simple example or application. The presentation should appeal to, and be readable by, students, especially students in engineering and science. Without being excessively theoretical, the book does address a number of unusual topics: Massera's theorem, Lyapunov's inequality, the isoperimetric inequality, numerical solutions of nonlinear boundary value problems, and more. There are also some new approaches to standard topics including a rethought presentation of series solutions and a nonstandard, but more intuitive, proof of the existence and uniqueness theorem. The collection of problems is especially rich and contains many very challenging exercises. Philip Korman is professor of mathematics at the University of Cincinnati. He is the author of over one hundred research articles in differential equations and the monograph Global Solution Curves for Semilinear Elliptic Equations. Korman has served on the editorial boards of Communications on Applied Nonlinear Analysis, Electronic Journal of Differential Equations, SIAM Review, an\ d Differential Equations and Applications.

Elliptic Partial Differential Equations

Elliptic Partial Differential Equations
Author: Qing Han
Publisher: American Mathematical Soc.
Total Pages: 161
Release: 2011
Genre: Mathematics
ISBN: 0821853139

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.

Index Theory of Elliptic Boundary Problems

Index Theory of Elliptic Boundary Problems
Author: Stephan Rempel
Publisher: Walter de Gruyter GmbH & Co KG
Total Pages: 396
Release: 1982-12-31
Genre: Mathematics
ISBN: 311270715X

No detailed description available for "Index Theory of Elliptic Boundary Problems".

Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains

Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains
Author: Mikhail S. Agranovich
Publisher: Springer
Total Pages: 343
Release: 2015-05-06
Genre: Mathematics
ISBN: 3319146483

This book, which is based on several courses of lectures given by the author at the Independent University of Moscow, is devoted to Sobolev-type spaces and boundary value problems for linear elliptic partial differential equations. Its main focus is on problems in non-smooth (Lipschitz) domains for strongly elliptic systems. The author, who is a prominent expert in the theory of linear partial differential equations, spectral theory and pseudodifferential operators, has included his own very recent findings in the present book. The book is well suited as a modern graduate textbook, utilizing a thorough and clear format that strikes a good balance between the choice of material and the style of exposition. It can be used both as an introduction to recent advances in elliptic equations and boundary value problems and as a valuable survey and reference work. It also includes a good deal of new and extremely useful material not available in standard textbooks to date. Graduate and post-graduate students, as well as specialists working in the fields of partial differential equations, functional analysis, operator theory and mathematical physics will find this book particularly valuable.