On the relaxation of integral functionals depending on the symmetrized gradient

Abstract

We prove results on the relaxation and weak* lower semicontinuity of integral functionals of the formover the space BD(Ω) of functions of bounded deformation or over the Temam-Strang space U(Ω) := u ∈ BD(Ω) : div u ∈ L 2 (Ω) , depending on the growth and shape of the integrand f . Such functionals are interesting for example in the study of Hencky plasticity and related models.

Relaxation for an optimal design problem inBD(Ω)

“…In the case where the energy density takes the form or (where stands for the deviator of the matrix given by ), and the total energy also includes a surface term, a first relaxation result was proved in [18]. We also refer to [39] for the relaxation in the case where there is no surface energy and to [33, 40, 42] for related models concerning evolutions and homogenization, among a wider list of contributions.…”

Relaxation for an optimal design problem in $BD(Ω)$

“…In the case where the energy density takes the form Eu 2 or E D u 2 + (div u) 2 (where A D stands for the deviator of the N × N matrix A given by A D := A− 1 N tr(A)I), and the total energy also includes a surface term, a first relaxation result was proved in [18]. We also refer to [39] for the relaxation in the case where there is no surface energy and to [42], [33], and [40] for related models concerning evolutions and homogenization, among a wider list of contributions.…”