Power-Law Approximation under Differential Constraints

Abstract

We study the Γ-convergence of the power-law functionals Fp(V) = Ω f p (x, V (x))dx 1/p , as p tends to +∞, in the setting of constant-rank operator A. We show that the Γ-limit is given by a supremal functional on L ∞ (Ω; M d×N) ∩ KerA where M d×N is the space of d × N real matrices. We give an explicit representation formula for the supremand function. We provide some examples and as application of the Γ-convergence results we characterize the strength set in the context of electrical resistivity.

Γ-convergence for power-law functionals with variable exponents

“…We conclude this section with the following lemma, already shown in [1] when f = f (x, ξ) is a Carathéodory integrand (see Lemma 4.5 therein). For the reader's convenience we report here the proof.…”

“…In [11] the authors consider the case when f (x, ξ) = a(x)|ξ| while in [1] the Γ-convergence is studied in the wider class of A ∞ -quasiconvex function f (see Definition 3.2 therein). These results have been extended in [10] in the setting of variable exponent Lebesgue space when f (x, •) is quasiconvex in the sense of Morrey.…”

A relaxation result in the vectorial setting and $L^p$-approximation for $L^\infty$-functionals

“…In [5] it has been introduced the class of functions f : R d×N → [0, +∞) satisfying f = lim p→∞ (Q(f p )) Proof of Theorem 1.2. The proof will be achieved in several steps, some of them follow along the lines of [5, Proof of Theorem 4.2].…”

“…In [25] the last assumption is dropped and the level convexity of the relaxed functional is proved for a class of discontinuos supremand, not even coercive. Despite of all these scalar results, very little is known in the vectorial setting, up to some sufficient conditions and in particular cases (see [5,6,13,19]).…”

A Γ-convergence result for optimal design problems

“…In the last two references, analogous asymptotic analyses using Γconvergence techniques for functionals involving single phase elastic density can be found. Also in [1,2], where the authors obtain limit models under some differential constraints, involving supremal functions and A -quasiconvex envelopes. We mention also [7] where the authors obtained an L p approximation and a lower semicontinuity result for supremal functionals.…”