Let u be an eigenfunction of the Laplacian on a compact mani-fold with boundary, with Dirichlet or Neumann boundary conditions, and let −λ 2 be the corresponding eigenvalue. We consider the problem of estimating max M u in terms of λ, for large λ, assuming M u 2 = 1. We prove that max M u ≤ C M λ (n−1)/2 , which is optimal for some M. Our proof simplifies… (More)
In this paper, continuing our earlier article [CGIKO], we study qualitative properties of solutions of a certain eigenvalue optimization problem. Especially we focus on the study of the free boundary of our optimal solutions on general domains.