Homogenization of Multiple Integrals

by Braides, Andrea; Defranceschi, Anneliese

• ISBN13: 9780198502463
• ISBN10: 019850246X
• Format: Hardcover
• Copyright: 1999-02-18
• Publisher: Clarendon Press

Summary

Homogenization theory describes the macroscopic properties of structures with fine microstructure. Its applications are diverse and include optimal design and the study of composites. The theory relies on the asymptotic analysis of fast-oscillating differential equations or integral functionals. This book is an introduction to the homogenization of nonlinear integral functionals. It emphasizes general results that do not rely on smoothness or convexity assumptions. The book presents a rigorous mathematical description of the overall properties of such functionals, with various applications that range from cellular elastic materials to Riemannian metrics and Hamilton-Jacobi equations. The book also includes self-contained introductions to the theories of gamma-convergence and weak lower semicontinuous functionals.

Table of Contents

Notation	xiii
Introduction	1	(10)
I LOWER SEMICONTINUITY OF INTEGRAL FUNCTIONALS	11	(54)
1 Lower semicontinuity and coerciveness	11	(7)
1.1 Lower semicontinuity	11	(2)
1.2 Yosida transforms	13	(3)
1.3 Coerciveness conditions. The direct method	16	(1)
1.4 Exercises	17	(1)
2 Weak convergence	18	(10)
2.1 Weak convergence in Lebesgue spaces	18	(4)
2.2 Weak convergence in Sobolev spaces	22	(1)
2.3 Weak* convergence of measures	22	(2)
2.4 Weak compactness criteria in L1	24	(3)
2.5 Exercises	27	(1)
3 Minimum problems in Sobolev spaces	28	(5)
3.1 The direct method. An example of application	28	(1)
3.2 Borel and Caratheodory functions	29	(2)
3.3 Rellich's Theorem and equivalent conditions for lower semicontinuity	31	(1)
3.4 Exercises	32	(1)
4 Necessary conditions for weak lower semicontinuity	33	(9)
4.1 General necessary conditions	33	(1)
4.2 W(1,p)-quasiconvexity	34	(6)
4.3 Rank-1-convexity	40	(1)
4.4 Exercises	41	(1)
5 Sufficient conditions for weak lower semicontinuity	42	(12)
5.1 Convexity	42	(3)
5.2 Polyconvexity	45	(3)
5.3 Quasiconvexity	48	(4)
5.4 Exercises	52	(2)
6 The structure of quasiconvex functions	54	(11)
6.1 Quasiconvexity of polyconvex functions	54	(1)
6.2 Quasiconvexification	55	(4)
6.3 Example of a quasiconvex non-polyconvex function	59	(1)
6.4 Example of a rank-1-convex non-quasiconvex function	60	(5)
II Gamma-CONVERGENCE	65	(36)
7 A naive introduction to Gamma-convergence	65	(8)
7.1 Definition and basic properties	65	(2)
7.2 Lower and upper Gamma-limits	67	(3)
7.3 Further properties. Compactness	70	(2)
7.4 Exercises	72	(1)
8 The indirect methods of Gamma-convergence	73	(4)
8.1 Gamma-limits and Yosida transforms	73	(1)
8.2 An example: Gamma-limits of quadratic functionals	74	(3)
9 Direct methods. An integral representation result	77	(5)
9.1 Localization	77	(1)
9.2 Integral representation on Sobolev spaces	77	(4)
9.3 Integral representation of homogeneous functionals	81	(1)
10 Increasing set functions	82	(3)
10.1 Increasing set functions	82	(1)
10.2 A characterization of measures as set functions	82	(2)
10.3 Increasing set functions and compactness of Gamma-limits	84	(1)
11 The fundamental estimate	85	(8)
11.1 Fundamental estimates	85	(3)
11.2 Subadditivity of Gamma-limits	88	(2)
11.3 Gamma-limits and boundary values	90	(2)
11.4 Exercises	92	(1)
12 Integral functionals with standard growth conditions	93	(8)
12.1 Standard growth conditions	93	(1)
12.2 Fundamental estimate	93	(2)
12.3 Compactness for the Gamma-limits	95	(1)
12.4 Gamma-limits of homogeneous functionals	96	(2)
12.5 Exercises	98	(3)
III BASIC HOMOGENIZATION	101	(66)
13 A 1-dimensional example	101	(7)
13.1 The cell-problem homogenization formula	101	(2)
13.2 The asymptotic homogenization formula	103	(1)
13.3 Proof of the Gamma-convergence	104	(2)
13.4 Exercises	106	(2)
14 Periodic homogenization	108	(20)
14.1 The asymptotic homogenization formula	109	(2)
14.2 The Homogenization Theorem	111	(3)
14.3 Convex homogenization	114	(6)
14.3.1 The cell-problem formula	114	(1)
14.3.2 Non-coercive convex homogenization	115	(5)
14.4 A counterexample to the cell-problem formula	120	(3)
14.5 An application: homogenization of elliptic equations in divergence form	123	(2)
14.6 Exercises	125	(3)
15 Almost-periodic homogenization	128	(14)
15.1 Homogenization of uniformly almost-periodic functionals	128	(7)
15.2 An example: loss of smoothness by homogenization	135	(5)
15.3 Exercises	140	(2)
16 Two applications	142	(8)
16.1 Homogenization of Riemannian metrics	142	(3)
16.2 Homogenization of Hamilton Jacobi equations	145	(5)
17 A closure theorem for the homogenization	150	(10)
17.1 A closure theorem	150	(6)
17.2 An application: homogenization of Besicovitch almost-periodic functionals	156	(4)
18 Loss of polyconvexity by homogenization	160	(7)
18.1 An example	160	(7)
IV FINER HOMOGENIZATION RESULTS	167	(106)
19 Homogenization of connected media	167	(14)
19.1 A homogenization theorem on periodic connected domains	167	(10)
19.2 Convergence of Neumann boundary value problems	177	(2)
19.3 Convergence of Dirichlet boundary value problems	179	(2)
20 Homogenization with stiff and soft inclusions	181	(18)
20.1 Media with stiff and soft inclusions	181	(2)
20.2 The Homogenization Theorem	183	(7)
20.3 Convergence of minima	190	(3)
20.4 A Lavrentiev phenomenon	193	(3)
20.5 Loss of polyconvexity after homogenization	196	(3)
21 Homogenization with non-standard growth conditions	199	(15)
21.1 A class of non-standard integrals	199	(3)
21.2 Convex homogenization	202	(1)
21.3 Non-convex homogenization	203	(9)
21.4 Exercises	212	(2)
22 Interated homogenization	214	(13)
22.1 Statement of the Iterated Homogenization Theorem	214	(1)
22.2 Proof of the Iterated Homogenization Theorem	215	(7)
22.3 Exercises	222	(5)
23 Correctors for homogenization	227	(22)
23.1 Convergence of momenta in homogenization	227	(7)
23.2 Definition and some properties of the correctors	234	(6)
23.3 Statement and proof of the correctors result	240	(6)
23.4 Correctors in the quasiperiodic case	246	(2)
23.5 Exercises	248	(1)
24 Homogenization of multi-dimensional structures	249	(24)
24.1 A smooth approach	249	(4)
24.2 A measure Sobolev-space approach	253	(10)
24.3 Homogenization of periodic thin structures	263	(5)
24.4 Exercises	268	(5)
V APPENDICES	273	(16)
A Almost-periodic functions	273	(4)
B Construction of extension operators	277	(10)
C Some regularity results	287	(2)
References	289	(8)
Notes to references	294	(3)
Index	297