Abstract
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 ...
Abstract
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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