Some Remarks on the Isoperimetric Problem for the Higher Eigenvalues of the Robin and Wentzell Laplacians
نویسنده
چکیده
We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in R of given volume M . When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant α in the boundary condition, or equivalently of the volume M . We obtain similar results for the Laplacian with generalised Wentzell boundary conditions ∆u+ β ∂u ∂ν + γu = 0.
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