D. Grieser

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We consider the following eigenvalue optimization problem: Given a bounded domain Ω ⊂ R n and numbers α ≥ 0, A ∈ [0, |Ω|], find a subset D ⊂ Ω of area A for which the first Dirichlet eigenvalue of the operator −∆ + αχD is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example, we show that for some(More)
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)
We consider the following eigenvalue optimization problem: Given a bounded domain ⊂ R and numbers α > 0, A ∈ [0, ||], find a subset D ⊂ of area A for which the first Dirichlet eigenvalue of the operator − + αχ D is as small as possible. We prove existence of solutions and investigate their qualitative properties. For example , we show that for some(More)
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