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.


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
Full Text: DOI arXiv


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