The average distance problem with perimeter-to-area ratio penalization

2022/01/25 by Qiang Du, Du, Qiang, Xin Yang Lu +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.2201.10100

arxiv created 2022/01/25 · arxiv updated 2022/01/26

Abstract

In this paper we consider the functional E_p,\la(Ω):=∫_Ω\dist^p(x,\pd Ω)\d x+\la (\H¹(\pd Ω))/(\H²(Ω)). Here p≥ 1, \la>0 are given parameters, the unknown Ω varies among compact, convex, Hausdorff two-dimensional sets of \R², \pd Ω denotes the boundary of Ω, and \dist(x,\pd Ω):=inf_{y∈\pd Ω}|x-y|. The integral term ∫_Ω\dist^p(x,\pd Ω)\d x quantifies the "easiness" for points in Ω to reach the boundary, while (\H¹(\pd Ω))/(\H²(Ω)) is the perimeter-to-area ratio. The main aim is to prove existence and C^1,1-regularity of minimizers of \E.

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