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E-raamat: Relaxation in Optimization Theory and Variational Calculus [De Gruyter e-raamatud]

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The series is devoted to the publication of high-level monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization, control theory, systems theory, mechanics, engineering, and other sciences. One of its main objectives is to make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications of nonlinear analysis. Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate further research at the crossroads of these areas are particularly welcome.

Editor-in-Chief Jürgen Appell, Würzburg, Germany

Honorary and Advisory Editors Catherine Bandle, Basel, Switzerland Alain Bensoussan, Richardson, Texas, USA Avner Friedman, Columbus, Ohio, USA Umberto Mosco, Worcester, Massachusetts, USA



Editorial Board Manuel del Pino, Bath, UK, and Santiago, Chile Mikio Kato, Nagano, Japan Wojciech Kryszewski, Toru, Poland Viceniu D. Rdulescu, Kraków, Poland Simeon Reich, Haifa, Israel

Please submit book proposals to Jürgen Appell.

Titles in planning include Ireneo Peral Alonso and Fernando Soria, Elliptic and Parabolic Equations Involving the HardyLeray Potential (2020) Cyril Tintarev, Profile Decompositions and Cocompactness: Functional-Analytic Theory of Concentration Compactness (2020) Takashi Suzuki, Semilinear Elliptic Equations: Classical and Modern Theories (2021)
Preface vii
1 Background generalities
1(67)
1.1 Order and topology
1(8)
1.2 Linear and convex analysis
9(7)
1.3 Optimization theory
16(19)
1.4 Function and measure spaces
35(12)
1.5 Means of continuous functions
47(3)
1.6 Some differential and integral equations
50(10)
1.7 Non-cooperative game theory
60(8)
2 Theory of convex compactifications
68(34)
2.1 Convex compactifications
69(2)
2.2 Canonical form of convex compactifications
71(10)
2.3 Convex (XXX)-compactifications
81(11)
2.4 Approximation of convex compactifications
92(4)
2.5 Extension of mappings
96(6)
3 Young measures and their generalizations
102(120)
3.1 Classical Young measures
103(17)
3.2 Various generalizations
120(31)
3.2.a Generalization by Fattorini
120(3)
3.2.b Generalization by Schonbek, Ball, Kinderlehrer and Pedregal
123(8)
3.2.c Generalization by DiPerna and Majda
131(17)
3.2.d Fonseca's extension of L(1)-spaces
148(3)
3.3 Convex compactifications of balls in L(p)-spaces
151(22)
3.4 Convex (XXX)-compactifications of L(p)-spaces
173(15)
3.5 Approximation theory
188(14)
3.6 Extensions of Nemytskii mappings
202(20)
3.6.a One-argument mappings: affine extensions
202(4)
3.6.b Two-argument mappings: semi-affine extensions
206(9)
3.6.c Two-argument mappings: bi-affine extensions
215(7)
4 Relaxation in optimization theory
222(98)
4.1 Abstract optimization problems
223(19)
4.2 Optimization problems on Lebesgue spaces
242(10)
4.3 Example: Optimal control of dynamical systems
252(35)
4.3.a Original problem
252(9)
4.3.b Relaxation scheme, correctness, well-posedness
261(6)
4.3.c Optimality conditions
267(8)
4.3.d Approximation theory
275(7)
4.3.e Sample calculations
282(5)
4.4 Example: Elliptic optimal control problems
287(13)
4.5 Example: Parabolic optimal control problems
300(12)
4.6 Example: Optimal control of integral equations
312(8)
5 Relaxation in variational calculus I
320(46)
5.1 Convex compactifications of Sobolev spaces
321(10)
5.2 Relaxation of variational problems; p > 1
331(8)
5.3 Optimality conditions for relaxed problems
339(10)
5.4 Relaxation of variational problems; p = 1
349(5)
5.5 Convex approximations of relaxed problems
354(12)
6 Relaxation in variational calculus II
366(46)
6.1 Prerequisities around quasiconvexity
367(5)
6.2 Gradient generalized Young functionals
372(11)
6.3 Relaxation scheme and its FEM-approximation
383(8)
6.4 Further approximation: an inner case
391(3)
6.5 Further approximation: an outer case
394(5)
6.6 Double-well problem: sample calculations
399(13)
7 Relaxation in game theory
412(31)
7.1 Abstract game-theoretical problems
412(7)
7.2 Games on Lebesgue spaces
419(4)
7.3 Example: Games with dynamical systems
423(14)
7.4 Example: Elliptic games
437(6)
Bibliography 443(20)
List of Symbols
463(6)
Index 469