Morrey spaces and generalized Cheeger sets

We maximize the functional\frac{\int_{E}h(x)\,dx}{P(E)},where {E\subset\overline{\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)}, {\lambda\geq n-1}, are natural spaces to study this problem. We prove that if {h\in 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 {h\in 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 {n\leq 7}). For the critical case {h\in 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 {h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h.