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Robin spectrum: two disks maximize the third eigenvalue. (English) Zbl 1482.35144

Summary: The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided the Robin parameter lies in a certain range and is scaled in each case by the length of the boundary. Equality is achieved when the domain degenerates suitably to the two disks.

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J25 Boundary value problems for second-order elliptic equations
30C70 Extremal problems for conformal and quasiconformal mappings, variational methods
74K15 Membranes
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