| Preface |
|
xi | |
| Preface to the second edition |
|
xviii | |
|
1 Background Generalities |
|
|
1 | (80) |
|
|
|
1 | (8) |
|
1.2 Linear, nonlinear, and convex analysis |
|
|
9 | (16) |
|
1.2.a Linear functional analysis |
|
|
10 | (4) |
|
|
|
14 | (4) |
|
1.2.c Means of continuous functions |
|
|
18 | (4) |
|
1.2.d Solving abstract nonlinear equations |
|
|
22 | (3) |
|
1.3 Function and measure spaces |
|
|
25 | (13) |
|
1.3.a Bochner and Lebesgue spaces |
|
|
26 | (4) |
|
|
|
30 | (3) |
|
1.3.c Spaces of smooth functions and Sobolev spaces |
|
|
33 | (5) |
|
1.4 Some differential and integral equations |
|
|
38 | (17) |
|
1.4.a Ordinary differential and differential-algebraic equations |
|
|
38 | (7) |
|
1.4.b Partial differential equations of elliptic type |
|
|
45 | (5) |
|
1.4.C Partial differential equations of parabolic type |
|
|
50 | (4) |
|
1.4.d Integral equations of Hammerstein type |
|
|
54 | (1) |
|
1.5 Basics from optimization theory |
|
|
55 | (26) |
|
1.5.a Existence, stability, approximation |
|
|
55 | (6) |
|
1.5.b Optimality conditions of the 1st order |
|
|
61 | (9) |
|
1.5.c Multicriteria optimization |
|
|
70 | (2) |
|
1.5.d Non-cooperative game theory |
|
|
72 | (9) |
|
2 Theory of Convex Compactifications |
|
|
81 | (36) |
|
2.1 Convex compactifications |
|
|
82 | (2) |
|
2.2 Canonical form of convex compactifications |
|
|
84 | (9) |
|
2.3 Convex a-compactifkations-- |
|
|
93 | (10) |
|
2.4 Approximation of convex compactifications |
|
|
103 | (3) |
|
2.5 Extension of mappings |
|
|
106 | (5) |
|
2.6 Inverse systems of convex compactifications |
|
|
111 | (6) |
|
3 Young Measures and Their Generalizations |
|
|
117 | (126) |
|
3.1 Classical Young measures |
|
|
118 | (17) |
|
3.1.a Basic scenario and results |
|
|
118 | (13) |
|
|
|
131 | (2) |
|
|
|
133 | (2) |
|
3.2 Various generalizations |
|
|
135 | (31) |
|
3.2.a Generalization by Fattorini |
|
|
136 | (2) |
|
3.2.b Generalization by Schonbek, Ball, Kinderlehrer and Pedregal |
|
|
138 | (8) |
|
3.2.c Generalization by DiPerna and Majda |
|
|
146 | (17) |
|
3.2.d Fonseca's extension of L1-spaces |
|
|
163 | (3) |
|
3.3 A class of convex compactifications of balls in Lp-spaces |
|
|
166 | (25) |
|
3.3.a Generalized Young functionals vpH,q(Ω S) |
|
|
166 | (8) |
|
3.3.b The composition h ·e;q |
|
|
174 | (2) |
|
3.3.c Some concrete examples |
|
|
176 | (10) |
|
3.3.d Coarse polynomial compactification by algebraic moments |
|
|
186 | (2) |
|
3.3.e Compatible systems of Young functionals on B(I;LP) |
|
|
188 | (3) |
|
3.4 A class of convex σ-compactifications of Lp-spaces |
|
|
191 | (13) |
|
|
|
204 | (18) |
|
3.5.a A general construction |
|
|
205 | (6) |
|
3.5.b An approximation over Ω |
|
|
211 | (4) |
|
3.5.c An approximation over 5 |
|
|
215 | (4) |
|
3.5.d Higher-order constructions by quasi-interpolatlon |
|
|
219 | (3) |
|
3.6 Extensions of Nemytskii mappings |
|
|
222 | (21) |
|
3.6.a One-argument mappings: affine extensions |
|
|
223 | (3) |
|
3.6.b Two-argument mappings: semi-affine extensions |
|
|
226 | (10) |
|
3.6.c Two-argument mappings: bi-affine extensions |
|
|
236 | (7) |
|
4 Relaxation in Optimization Theory |
|
|
243 | (139) |
|
4.1 Abstract optimization problems |
|
|
244 | (19) |
|
4.2 Optimization problems on Lebesgue spaces |
|
|
263 | (14) |
|
4.3 Optimal control of finite-dimensional dynamical systems |
|
|
277 | (48) |
|
|
|
277 | (11) |
|
4.3.b Relaxation scheme, correctness, well-posedness |
|
|
288 | (7) |
|
4.3.C Optimality conditions |
|
|
295 | (10) |
|
4.3.d Approximation theory |
|
|
305 | (5) |
|
4.3.e Illustrative computational simulations: oscillations |
|
|
310 | (5) |
|
4.3.f Illustrative computational simulations: oscillations and concentrations |
|
|
315 | (3) |
|
4.3.g Optimal control of differential-algebraic systems |
|
|
318 | (7) |
|
4.4 Elliptic optimal control problems |
|
|
325 | (21) |
|
4.4.a The original problem and its relaxation |
|
|
325 | (8) |
|
4.4.b Optimality conditions in semilinear case |
|
|
333 | (6) |
|
4.4.c Optimal control of Navier-Stokes' equations |
|
|
339 | (3) |
|
4.4.d Optimal material design of some stratified media |
|
|
342 | (4) |
|
4.5 Parabolic optimal control problems |
|
|
346 | (27) |
|
4.5.a Infinite-dimensional dynamical-system approach |
|
|
348 | (7) |
|
4.5.b An approach through parabolic partial differential equations |
|
|
355 | (10) |
|
4.5.C Optimal control of Navier-Stokes equations |
|
|
365 | (8) |
|
4.6 Optimal control of integral equations |
|
|
373 | (9) |
|
5 Relaxation in Variational Calculus: Scalar Case |
|
|
382 | (53) |
|
5.1 Convex compactifications of Sobolev spaces |
|
|
383 | (11) |
|
5.2 Relaxation of variational problems; p < 1 |
|
|
394 | (7) |
|
5.3 Optimality conditions for relaxed problems |
|
|
401 | (9) |
|
5.4 Relaxation of variational problems; p = 1 |
|
|
410 | (6) |
|
5.5 Convex approximations of relaxed problems |
|
|
416 | (13) |
|
5.6 Example: Mlcrostructure in ferromagnetic materials |
|
|
429 | (6) |
|
6 Relaxation In Variational Calculus: Vectorial Case |
|
|
435 | (48) |
|
6.1 Prerequisites around quasiconvexity |
|
|
436 | (5) |
|
6.2 Gradient generalized Young functionals |
|
|
441 | (12) |
|
6.3 Variational problems and their relaxation |
|
|
453 | (5) |
|
|
|
458 | (3) |
|
6.5 Further approximation: an inner case |
|
|
461 | (3) |
|
6.6 Further approximation: an outer case |
|
|
464 | (5) |
|
6.7 Multiwell problems: illustrative calculations |
|
|
469 | (14) |
|
7 Relaxation in Game Theory |
|
|
483 | (30) |
|
7.1 Abstract game-theoretical problems |
|
|
484 | (6) |
|
7.2 Games on Lebesgue spaces |
|
|
490 | (3) |
|
7.3 Example: Games with dynamical systems |
|
|
493 | (14) |
|
7.4 Example: Elliptic games |
|
|
507 | (6) |
|
8 Relaxation in evolutionary problems |
|
|
513 | (26) |
|
8.1 Evolution on abstract convex compactifications |
|
|
513 | (10) |
|
8.1.a Rate-independent evolution |
|
|
515 | (6) |
|
8.1.b Quasistatic rate-dependent evolution |
|
|
521 | (2) |
|
8.2 Applications of relaxation in rate-independent evolution |
|
|
523 | (9) |
|
8.2.a Perfect plasticity at small strains |
|
|
524 | (2) |
|
8.2.b Evolution of microstructure in ferromagnetic materials |
|
|
526 | (2) |
|
8.2.c Evolution of microstructure in shape-memory materials |
|
|
528 | (4) |
|
8.3 Notes about measure-valued solutions to parabolic equations |
|
|
532 | (7) |
| Bibliography |
|
539 | (32) |
| List of Symbols |
|
571 | (6) |
| Index |
|
577 | |
