Calculus of Variations

Abstract

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A note on non-homogeneous deformations with homogeneous Cauchy stress for a strictly rank-one convex energy in isotropic hyperelasticity

“…Recall that in order to satisfy the jump condition, the deformation gradients along the interface must be rankone connected [7]. However, it can be shown [14,Lemma 8.25] that rank(F 1 −F 2 ) = 1 for all F 1 , F 2 ∈ CSO(2); note that each F ∈ CSO(2) is of the form…”

On the relaxation of integral functionals depending on the symmetrized gradient

“…If the function f additionally satisfies an asymptotic growth condition of the form |f (A)| ≤ C(1+|A| p ), p ∈ [1, ∞), then it is sufficient to test the above inequality with ψ ∈ W 1,p 0 (D; R d ) instead of Lipschitz functions (the proof is analogous to Lemma 7.1 in [33], also cf. Proposition 3.4 in [19]).…”

“…See also Proposition 5.11 in [15] for a different proof. (2) For a non-negative continuous function f with p-growth, 1 ≤ p < ∞, the symmetric-quasiconvex envelope SQf is symmetric-quasiconvex and also has p-growth (see Lemma 7.1 in [33]).…”

Dimensional estimates and rectifiability for measures satisfying linear PDE constraints

“…It has been shown in [16], see also [12,15] for recent surveys and [29,Chapter 10] for further explanation, that by suitably choosing the operator A, the study of the singular part of A-free measures has several consequences in the calculus of variations and in geometric measure theory. In particular, we recall the following:…”