Functional Analysis, Sobolev Spaces, and Calculus of Variations

Pablo Pedregal (Autor)

Buch | Softcover

XIV, 387 Seiten

2024
Springer International Publishing (Verlag)
978-3-031-49245-7 (ISBN)

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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.

lt;b>Pablo Pedregal received his Ph.D. degree in Mathematics from the University of Minnesota at the end of 1989, under the guidance of D. Kinderlehrer. Soon after that, he became Associate Professor at U. Complutense. During the academic year 1994-95 he moved to U. de Castilla-La Mancha where he led the Math Department for several years. In 1997 he became full professor. His field of interest focuses on variational techniques applied to optimization in a broad sense, including, but not limited to, calculus of variations-especially vector, non-convex problems-, optimal design in continuous media, optimal control, etc, and more recently he has explored variational ideas in areas like inverse problems, and dynamical systems, as well as optimal control problems governed by hyper-elasticity. He has regularly traveled to research centers in the USA and Europe. He has written more than one hundred research articles, seven specialized books (at least three of them with Springer), and some other of a more didactic style for undergraduates.

1 Motivation and perspective.- Part I: Basic Functional Analysis and Calculus of Variations.- 2 A first exposure to Functional Analysis.- 3 Introduction to convex analysis. The Hahn-Banach and Lax-Milgram theorems.- 4 The Calculus of Variations for one-dimensional problems.- Part II: Basic Operator Theory.- 5 Continuous operators.- 6 Compact operators.- Part III: Multidimensional Sobolev Spaces and Scalar Variational Problems.- 7 Multidimensional Sobolev spaces.- 8 Variational problems.- 9 Finer results in Sobolev spaces and the Calculus of Variations.- Appendix A: Hints and solutions to exercises.- Appendix B: So much to learn.

"The book is nicely written and the author's 'intentionally didactic' style certainly conveys a good amount of enthusiasm. ... The text offers an engaging and enthusiastically written introduction to the modern calculus of variations which may well prove to be a valuable resource to students and non-specialist researchers in the field." (David Seifert, zbMATH 1544.46001, 2024)

Erscheinungsdatum	01.02.2024
Reihe/Serie	La Matematica per il 3+2
Reihe/Serie	UNITEXT
Zusatzinfo	XIV, 387 p.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	756 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
Themenwelt	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
Schlagworte	adjoint operator • Banach space • Banach-Steinhauss principle • closed-graph theorem • Compact operator • Convex Analysis • Direct Method • Fourier transform • Hahn-Banach theorem • hilbert space • Lax-Milgram lemma • Lebesgue Space • open-mapping theorem • Optimality conditions • Poincaré's inequality • Sobolev inequalities • Sobolev Space • weak lower semicontinuity • Weak topology
ISBN-10	3-031-49245-5 / 3031492455
ISBN-13	978-3-031-49245-7 / 9783031492457
Zustand	Neuware