Density of polyhedral partitions

TL;DR

This paper demonstrates that finite Caccioppoli partitions can be approximated arbitrarily closely by polyhedral partitions through small deformations, facilitating simplified energy computations in variational problems.

Contribution

It proves the density of polyhedral partitions in the space of finite Caccioppoli partitions using smooth deformations, enabling easier analysis of partition-based energies.

Findings

Any finite Caccioppoli partition can be approximated by a polyhedral partition within any small error.

The approximation preserves energy estimates for a broad class of energies on partitions.

Such approximations are useful for simplifying calculations in $ ext{Gamma}$-convergence analysis.

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 $SBV_{\rm loc}(\Omega;{\cal Z})$ with ${\cal Z}\subset \mathbb R^N$ a finite set of parameters. For all $\varepsilon>0$ we prove that such a $u$ is $\varepsilon$-close to a small deformation of a polyhedral decomposition $v_\varepsilon$, in the sense that there is a $C^1$ diffeomorphism $f_\varepsilon:\mathbb R^n\to\mathbb R^n$ which is $\varepsilon$-close to the identity and such that $u\circ f_\varepsilon-v_\varepsilon$ is $\varepsilon$-small in the strong $BV$ norm. This implies that the energy of $u$ is close to that of $v_\varepsilon$ 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.