Kreisbeck, Carolin and Rindler, Filip (2015) Thin-Film Limits of Functionals on A-free Vector Fields. INDIANA UNIVERSITY MATHEMATICS JOURNAL, 64 (5). pp. 1383-1423. ISSN 0022-2518, 1943-5258
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This paper deals with variational principles on thin films subject to linear PDE constraints represented by a constant-rank operator A. We study the effective behavior of integral functionals as the 'thickness of the domain tends to zero, investigating both upper and lower bounds for the Gamma-limit. Under certain conditions, we show that the limit is an integral functional, and we give an explicit formula. The limit functional turns out to be constrained to A(0)-free vector fields, where the limit operator A is in general not of constant rank. This result extends work by Bouchitte, Fonseca, and Mascarenhas [J Convex Anal. 16 (2009), pp. 351-365] to the setting of A-free vector fields. While the lower bound follows from a Young measure approach together with a new decomposition lemma, the construction of a recovery sequence relies on algebraic considerations in Fourier space. This part of the argument requires a careful analysis of the limiting behavior of the resealed operators A(epsilon) by a suitable convergence of their symbols, as well as an explicit construction for plane waves inspired by the bending moment formulas in the theory of (linear) elasticity. We also give a few applications to common operators A.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | DIMENSION REDUCTION PROBLEMS; NONLINEAR ELASTICITY; LOWER SEMICONTINUITY; EQUI-INTEGRABILITY; MEMBRANE THEORY; CONVERGENCE; QUASICONVEXITY; HOMOGENIZATION; RELAXATION; INTEGRALS; Dimension reduction; thin films; PDE constraints; A-quasiconvexity; Gamma-convergence |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 26 Jul 2019 09:53 |
| Last Modified: | 26 Jul 2019 09:53 |
| URI: | https://pred.uni-regensburg.de/id/eprint/6154 |
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