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- W2633871002 abstract "Abstract We maximize the functional frac{int_{E}h(x),dx}{P(E)}, where {Esubsetoverline{Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω . We show that the Morrey spaces {L^{1,lambda}(Omega)} , {lambdageq n-1} , are natural spaces to study this problem. We prove that if {hin L^{1,lambda}(Omega)} , {lambda>n-1} , then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into {L^{p}} spaces. We show that, for any {0<lambda<n} , the Morrey space {L^{1,lambda}(Omega)} is not contained in any {L^{q}(Omega)} , {1<q<p=frac{n}{n-lambda}} . We also show that if {hin L^{1,lambda}(Omega)} , {lambda>n-1} , then the reduced boundary in Ω of a generalized Cheeger set is {C^{1,alpha}} and the singular set has Hausdorff dimension at most {n-8} (empty if {nleq 7} ). For the critical case {hin L^{1,n-1}(Omega)} , we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in {mathbb{R}^{n}} with {hin L^{1}(mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h ." @default.
- W2633871002 created "2017-06-30" @default.
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- W2633871002 date "2019-04-01" @default.
- W2633871002 modified "2023-10-17" @default.
- W2633871002 title "Morrey spaces and generalized Cheeger sets" @default.
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- W2633871002 doi "https://doi.org/10.1515/acv-2016-0050" @default.
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