Calculus of Variations and Geometric Measure Theory

A. Arroyo-Rabasa

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

created by arroyorabasa on 09 Jan 2017


Published Paper

Inserted: 9 jan 2017
Last Updated: 9 jan 2017

Journal: Calculus of Variations and Partial Differential Equations
Volume: 55
Number: 6
Pages: 154-197
Year: 2016
Doi: 10.1007/s00526-016-1085-5


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

\[(u, A) \mapsto \int_\Omega 2 f u \, \text{d}x − \int_{\Omega \cap A} \sigma_1\mathcal A u · \mathcal A u \, \text{d}x − \int_{\Omega \setminus A} \sigma_2 \mathcal A u · \mathcal A u \, \text{d}x + \text{Per}(A;\overline\Omega), \] 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 $C^1$-hypersurface up to a closed set of zero $\mathcal H^{N−1}$-measure.

Keywords: partial regularity, almost perimeter minimizer, optimal design problem