In this survey we deal with shape optimization problems involving convex combinations of the first two eigenvalues of the Dirichlet Laplacian, mainly recalling and explaining some recent results. More precisely, we discuss some geometric properties of minimizers, in particular when they are no longer convex and the optimality of balls. This leads us to deal with the "attainable set" of the first two eigenvalues, which is a great source of open problems.
Mazzoleni, D. C. S., Some remarks on convex combinations of low eigenvalues, (Torino, 02-05 May 2016), <<RENDICONTI DEL SEMINARIO MATEMATICO>>, 2016; 74 (3-4): 43-52 [http://hdl.handle.net/10807/117120]
Some remarks on convex combinations of low eigenvalues
Mazzoleni, Dario Cesare Severo
2016
Abstract
In this survey we deal with shape optimization problems involving convex combinations of the first two eigenvalues of the Dirichlet Laplacian, mainly recalling and explaining some recent results. More precisely, we discuss some geometric properties of minimizers, in particular when they are no longer convex and the optimality of balls. This leads us to deal with the "attainable set" of the first two eigenvalues, which is a great source of open problems.
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