05011nam a22004695i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001600172072001600188072002300204072001500227100008500242245011300327250001800440264006900458300003400527336002600561337002600587338003600613347002400649490004800673505065400721520271401375650003404089650003804123650003504161650001404196650005204210710003404262773002604296776003604322776003604358830005504394856004604449912001404495912001504509999001704524978-3-031-49246-4DE-He21320251030111407.0cr nn 008mamaa240130s2024    sz |    s    |||| 0|eng d  a97830314924649978-3-031-49246-47 a10.1007/978-3-031-49246-42doi 4aQA299.6-433 7aPBK2bicssc 7aMAT0340002bisacsh 7aPBK2thema1 aPedregal, Pablo.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut97014410aFunctional Analysis, Sobolev Spaces, and Calculus of Variationsh[electronic resource] /cby Pablo Pedregal.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXIV, 387 p.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aLa Matematica per il 3+2,x2038-5757 ;v1570 a1 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.  aThis 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. 0aMathematical analysis.964178 0aMathematical optimization.965522 0aCalculus of variations.96726814aAnalysis.24aCalculus of Variations and Optimization.9672702 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303149245708iPrinted edition:z9783031492471 0aLa Matematica per il 3+2,x2038-5757 ;v15797014540uhttps://doi.org/10.1007/978-3-031-49246-4  aZDB-2-SMA  aZDB-2-SXMS  c57370d57370