Integral representation for a relaxed optimal design problem for non-simple grade two materials
Ana Cristina Barroso, Elvira Zappale
TL;DR
The paper addresses the relaxation and integral representation of a relaxed design energy for two-phase materials with a bulk term depending on the second gradient and a perimeter penalty, posed in the BH and BV framework. It introduces the energy $F(\chi,u;\Omega)$ with $\chi\in BV(\Omega;{0,1})$ and $u\in BH(\Omega;\mathbb{R}^d)$ and proves that its lower semicontinuous envelope admits an exact Radon measure representation involving the $2$-quasiconvex envelope $Q^2f$ in the bulk, a Cantor term via $(Q^2f)^{\infty}$, and a jump term with density $g$ computed through a localized relaxation. The main result expresses the relaxed energy as $\mathcal{F}(\chi,u;A)=\int_A Q^2f(\chi, \nabla^2 u)\,dx+ \int_{A\cap(S_\chi\cup S_{\nabla u})} g(\chi^+,\chi^-, (\nabla u)^+, (\nabla u)^-, \nu) \, d\mathcal{H}^{N-1} + \int_A (Q^2f)^{\infty}(\chi, \frac{d D^c(\nabla u)}{d|D^c(\nabla u)|}) \, d|D^c(\nabla u)|$. This extends BV relaxation to the BH setting for non-simple materials and provides a rigorous foundation for relaxed optimal design in two-phase systems, including detailed treatment of bulk, Cantor, and jump contributions and conditions under which the surface density $g$ can be characterized sequentially. The results have implications for mathematical modeling of plastic deformations and structured deformations with linear growth.
Abstract
A measure representation result for a functional modelling optimal design problems for plastic deformations, under linear growth conditions, is obtained. Departing from an energy with a bulk term depending on the deformation gradient and its derivatives, as well as a perimeter term, the functional in question corresponds to the relaxation of this energy with respect to a pair $(χ,u)$, where $χ$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded hessian.