Thin-Film Limits of Functionals on A-free Vector Fields

被引:4
|
作者
Kreisbeck, Carolin [1 ]
Rindler, Filip [2 ]
机构
[1] Univ Regensburg, Fak Math, D-93040 Regensburg, Germany
[2] Univ Warwick, Math Inst, Coventry CV4 7AL, W Midlands, England
基金
英国工程与自然科学研究理事会;
关键词
Dimension reduction; thin films; PDE constraints; A-quasiconvexity; Gamma-convergence; NONLINEAR ELASTICITY; LOWER SEMICONTINUITY; EQUI-INTEGRABILITY; QUASICONVEXITY; CONVERGENCE; RELAXATION; ENERGY; MODELS;
D O I
10.1512/iumj.2015.64.5653
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
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页码:1383 / 1423
页数:41
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