Tag: Sobolev

Free Download Cornelia Schneider, "Beyond Sobolev and Besov: Regularity of Solutions of PDEs and Their Traces in Function Spaces "
English | ISBN: 3030751384 | 2021 | 348 pages | PDF | 3 MB
This book investigates the close relation between quite sophisticated function spaces, the regularity of solutions of partial differential equations (PDEs) in these spaces and the link with the numerical solution of such PDEs. It consists of three parts. Part I, the introduction, provides a quick guide to function spaces and the general concepts needed. Part II is the heart of the monograph and deals with the regularity of solutions in Besov and fractional Sobolev spaces. In particular, it studies regularity estimates of PDEs of elliptic, parabolic and hyperbolic type on non smooth domains. Linear as well as nonlinear equations are considered and special attention is paid to PDEs of parabolic type. For the classes of PDEs investigated a justification is given for the use of adaptive numerical schemes. Finally, the last part has a slightly different focus and is concerned with traces in several function spaces such as Besov- and Triebel-Lizorkin spaces, but also in quite general smoothness Morrey spaces.

Free Download Functional Analysis, Sobolev Spaces, and Calculus of Variations (UNITEXT, 157) by Pablo Pedregal
English | January 31, 2024 | ISBN: 3031492455 | 401 pages | MOBI | 51 Mb
This book aims at introducing students into the modern analytical foundations to treat problems and situations in the Calculus of Variations solidly and rigorously. Since no background is taken for granted or assumed, as the textbook pretends to be self-contained, areas like basic Functional Analysis and Sobolev spaces are studied to the point that chapters devoted to these topics can be utilized by themselves as an introduction to these important parts of Analysis. The material in this regard has been selected to serve the needs of classical variational problems, leaving broader treatments for more advanced and specialized courses in those areas. It should not be forgotten that problems in the Calculus of Variations historically played a crucial role in pushing Functional Analysis as a discipline on its own right. The style is intentionally didactic. After a first general chapter to place optimization problems in infinite-dimensional spaces in perspective, the first part of the book focuses on the initial important concepts in Functional Analysis and introduces Sobolev spaces in dimension one as a preliminary, simpler case (much in the same way as in the successful book of H. Brezis). Once the analytical framework is covered, one-dimensional variational problems are examined in detail including numerous examples and exercises. The second part dwells, again as a first-round, on another important chapter of Functional Analysis that students should be exposed to, and that eventually will find some applications in subsequent chapters. The first chapter of this part examines continuous operators and the important principles associated with mappings between functional spaces; and another one focuses on compact operators and their fundamental and remarkable properties for Analysis. Finally, the third part advances to multi-dimensional Sobolev spaces and the corresponding problems in the Calculus of Variations. In this setting, problems become much more involved and, for this same reason, much more interesting and appealing. In particular, the final chapter dives into a number of advanced topics, some of which reflect a personal taste. Other possibilities stressing other kinds of problems are possible. In summary, the text pretends to help students with their first exposure to the modern calculus of variations and the analytical foundation associated with it. In particular, it covers an extended introduction to basic functional analysis and to Sobolev spaces. The tone of the text and the set of proposed exercises will facilitate progressive understanding until the need for further challenges beyond the topics addressed here will push students to more advanced horizons.

Free Download Blow-up in Nonlinear Sobolev Type Equations By Alexander B. Al’shin
2011 | 648 Pages | ISBN: 3110255278 | PDF | 4 MB
The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time. The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.

Free Download Functional Analysis, Sobolev Spaces, and Calculus of Variations
English | 2024 | ISBN: 3031492455 | 391 Pages | PDF EPUB (True) | 27 MB
This book aims at introducing students into the modern analytical foundations to treat problems and situations in the Calculus of Variations solidly and rigorously. Since no background is taken for granted or assumed, as the textbook pretends to be self-contained, areas like basic Functional Analysis and Sobolev spaces are studied to the point that chapters devoted to these topics can be utilized by themselves as an introduction to these important parts of Analysis. The material in this regard has been selected to serve the needs of classical variational problems, leaving broader treatments for more advanced and specialized courses in those areas. It should not be forgotten that problems in the Calculus of Variations historically played a crucial role in pushing Functional Analysis as a discipline on its own right. The style is intentionally didactic. After a first general chapter to place optimization problems in infinite-dimensional spaces in perspective, the first part of the book focuses on the initial important concepts in Functional Analysis and introduces Sobolev spaces in dimension one as a preliminary, simpler case (much in the same way as in the successful book of H. Brezis). Once the analytical framework is covered, one-dimensional variational problems are examined in detail including numerous examples and exercises. The second part dwells, again as a first-round, on another important chapter of Functional Analysis that students should be exposed to, and that eventually will find some applications in subsequent chapters. The first chapter of this part examines continuous operators and the important principles associated with mappings between functional spaces; and another one focuses on compact operators and their fundamental and remarkable properties for Analysis. Finally, the third part advances to multi-dimensional Sobolev spaces and the corresponding problems in the Calculus of Variations. In this setting, problems become much more involved and, for this same reason, much more interesting and appealing. In particular, the final chapter dives into a number of advanced topics, some of which reflect a personal taste. Other possibilities stressing other kinds of problems are possible. In summary, the text pretends to help students with their first exposure to the modern calculus of variations and the analytical foundation associated with it. In particular, it covers an extended introduction to basic functional analysis and to Sobolev spaces. The tone of the text and the set of proposed exercises will facilitate progressive understanding until the need for further challenges beyond the topics addressed here will push students to more advanced horizons.

Free Download An Introduction to Sobolev Spaces
by Erhan Pişkin and Baver Okutmuştur
English | 2021 | ISBN: 1681089149 | 203 Pages | True PDF | 8.1 MB