Calculus of Variations and Geometric Measure Theory

V. Crismale - G. Orlando

A lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,γ}$, $γ>1$

created by orlando on 23 Sep 2019
modified by crismale on 13 Dec 2019


Published Paper

Inserted: 23 sep 2019
Last Updated: 13 dec 2019

Journal: Mathematics in Engineering
Volume: 2
Number: 1
Pages: 101-118
Year: 2020
Doi: 10.3934/mine.2020006

ArXiv: 1909.09615 PDF


We prove the lower semicontinuity of functionals of the form \[ \int \limits_\Omega \! V(\alpha) \, \mathrm{d}
\mathrm{E} u
\, , \] with respect to the weak converge of $\alpha$ in $W^{1,\gamma}(\Omega)$, $\gamma > 1$, and the weak$^*$ convergence of $u$ in $BD(\Omega)$, where $\Omega \subset \mathbb{R}^n$. These functional arise in the variational modelling of linearised elasto-plasticity coupled with damage and their lower semicontinuity is crucial in the proof of existence of quasi-static evolutions. This is the first result achieved for subcritical exponents $\gamma < n$.