Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

In this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

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.

On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells.