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Density of polyhedral partitions. (English) Zbl 1366.49051

Summary: We prove the density of polyhedral partitions in the set of finite Caccioppoli partitions. Precisely, given a decomposition \(u\) of a bounded Lipschitz set \(\Omega \subset {\mathbb {R}}^n\) into finitely many subsets of finite perimeter and \({\varepsilon }>0\), we prove that \(u\) is \({\varepsilon }\)-close to a small deformation of a polyhedral decomposition \(v_{\varepsilon }\), in the sense that there is a \(C^1\) diffeomorphism \(f_{\varepsilon }:{\mathbb {R}}^n\rightarrow {\mathbb {R}}^n\) 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.

MSC:

49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
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