Density of polyhedral partitions

Abstract

We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, we consider a decomposition $u$ of a bounded Lipschitz set $\Omega \subset {\mathbb{R}}^n$ into finitely many subsets of finite perimeter, which can be identified with a function in $SB{V}_{\mathrm{loc}}(\Omega ;\mathcal{Z})$ with $\mathcal{Z}\subset {\mathbb{R}}^N$ a finite set of parameters. For all $\epsilon >0$ we prove that such a $u$ is $\epsilon$-close to a small deformation of a polyhedral decomposition $v_{\epsilon }$, in the sense that there is a $C^1$ diffeomorphism $f_{\epsilon }:{\mathbb{R}}^n\to {\mathbb{R}}^n$ which is $\epsilon$-close to the identity and such that $u\circ f_{\epsilon }-v_{\epsilon }$ is $\epsilon$-small in the strong $BV$ norm. This implies that the energy of $u$ is close to that of $v_{\epsilon }$ for a large class of energies defined on partitions. Such type of approximations are very useful in order to simplify computations in the estimates of $\Gamma$-limits.

References in corpus (1)

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