Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains (vol 9, pg 857, 2019)

For Omega subset of R-n, a convex and bounded domain, we study the spectrum of -Delta(Omega) the Dirichlet Laplacian on Omega. For Lambda >= 0 and gamma >= 0 let Omega(Lambda,gamma)(A) denote any extremal set of the shape optimization problem sup {Tr(-Delta(Omega) - Lambda)(gamma) : Omega is an element of .A, vertical bar Omega vertical bar = 1}, where A is an admissible family of convex domains in R-n. If gamma >= 1 and (1) and {Lambda(j)}(j >= 1) is a positive sequence tending to infinity we prove that {Omega Lambda j , gamma(A)}(j >= 1) is a bounded sequence, and hence contains a convergent subsequence. Under an additional assumption on A we characterize the possible limits of such subsequences asminimizers of the perimeter among domains in A of unit measure. For instance if A is the set of all convex polygons with no more than m faces, then Omega(Lambda,gamma) converges, up to rotation and translation, to the regular m-gon.