Korn-Poincare inequalities for functions with a small jump set

Abstract

Functions in SBDp arise naturally in the study of geometrically linear fracture models. They have a jump set of finite (n − 1)-dimensional measure and, away from the jump set, a symmetrized gradient e(u) = (∇u + ∇u T )/2 in L p , p ≥ 1. We show that if the measure of the jump set is sufficiently small with respect to the size of the domain, then the function u can be approximated by an affine function away from a small exceptional set, with an error which depends solely on e(u). We also derive a corresponding trace statement.

Functionals Defined on Piecewise Rigid Functions: Integral Representation and $$\varGamma $$-Convergence

“…However, our primary purpose comes from the study of free-discontinuity problems defined on the space GSBD p , see [38], which has obtained steadily increasing attention over the last years, cf., e.g., [30,31,32,33,34,35,46,47,48,50]. We have indeed already mentioned before how the analysis of partition problems has proved to be a relevant tool in the study of freediscontinuity problems on SBV .…”

Existence of strong solutions to the Dirichlet problem for the Griffith energy

“…If v ∈ GSBD(U ), with e(v) ∈ L p (U ; M n×n sym ), p > 1, and H n−1 (J v ) < ∞, then v ∈ GSBD p (U ). The following result has been proven by Chambolle, Conti, and Francfort in [12], stated in SBD p . The proof, only based on one dimensional slicing, holds in fact for functions in GSBD p , and this has been employed for instance in [14,15].…”

“…This passes also through an approximation of GSBD displacements with small jump set and a prescribed value in a subdomain D, through functions keeping the same value both in D and near the boundary, and smooth in the interior (Theorem 3.1). The proof of the compactness result is done in the spirit of the corresponding [14,Theorem 3], employing a Korn-Poincaré-type estimate by [12]. The regularity of minimisers with prescribed value on a subdomain is obtained in Theorem 2.6 by adapting a regularity result for solution to elliptic systems in [39].…”

Equilibrium Configurations for Epitaxially Strained Films and Material Voids in Three-Dimensional Linear Elasticity

“…We now proceed with the proof of Lemma 6.5 which relies strongly on [15, Theorem 3.1]. Another main ingredient is the following Korn-Poincaré inequality in GSBD p , see [13,Proposition 3]. The proof of (3.1a), (3.1d), (3.1b) in [15, Theorem 3.1] may be followed exactly, with the modifications described just above and the suitable slight change of notation.…”

“…Proof. We refer to the first step in the proof of [18, Proposition 4.1], in particular [18, Equation (12)- (13)]. We point out that the case of possibly unbounded graphs has been treated there, i.e., h ∈ BV (ω; [0, +∞)).…”