Calculus of Variations and Geometric Measure Theory

A. Braides - S. Conti - A. Garroni

Density of polyhedral partitions

created by braidesa on 12 Sep 2016
modified by garroni on 18 Sep 2020


Published Paper

Inserted: 12 sep 2016
Last Updated: 18 sep 2020

Journal: Calc. Var. Partial Diff. Equations
Volume: 56
Number: 2
Pages: 10
Year: 2017
Doi: 10.1007/s00526-017-1108-x


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 types of approximations are very useful in order to simplify computations in the estimates of $\Gamma$-limits.