Density of polyhedral partitions

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• 作者：Andrea Braides ; Sergio Conti…
• 关键词：Mathematics Subject Classification49Q15 ; 49Q20 ; 49J45
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：56
• 期：2
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-0835
• 卷排序：56

文摘

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.