Regularity for free interface variational problems in a general class of gradients

Abstract

We present a way to study a wide class of optimal design problems with a perimeter penalization. More precisely, we address existence and regularity properties of saddle points of energies of the form $$(u,A)\mapsto {\int }_{\Omega }2fu\text{d}x-{\int }_{\Omega \cap A}{\sigma }_1\mathcal{A}u\cdot \mathcal{A}u\text{d}x-{\int }_{\Omega \setminus A}{\sigma }_2\mathcal{A}u\cdot \mathcal{A}u\text{d}x+\text{Per}(A;\overline{\Omega }),$$ where $\Omega$ is a bounded Lipschitz domain, $A\subset {\mathbb{R}}^N$ is a Borel set, $u:\Omega \subset {\mathbb{R}}^N\to {\mathbb{R}}^d$, $\mathcal{A}$ is an operator of gradient form, and ${\sigma }_1,{\sigma }_2$ are two not necessarily well-ordered symmetric tensors. The class of operators of gradient form includes scalar- and vector-valued gradients, symmetrized gradients, and higher order gradients. Therefore, our results may be applied to a wide range of problems in elasticity, conductivity or plasticity models. In this context and under mild assumptions on $f$, we show for a solution $(w,A)$, that the topological boundary of $A\cap \Omega$ is locally a ${\mathrm{C}}^1$-hypersurface up to a closed set of zero ${\mathcal{H}}^{N-1}$-measure.

arXiv commentThe final publication is available at Springer via http://dx.doi.org/10.1007/s00526-016-1085-5