## Some remarks on the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

### JB Kennedy

#### Abstract

We consider the problem of minimising the $$k$$th eigenvalue, $$k \ge 2$$, of the ($$p$$-)Laplacian with Robin boundary conditions with respect to all domains in $$\mathbb{R}^N$$ 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 $$k \ge 3$$, we prove that in many cases a minimiser cannot be independent of the value of the constant $$\alpha$$ in the boundary condition, or equivalently of the volume $$M$$. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions $$\Delta u + \beta \frac{\partial u}{\partial \nu} + \gamma u = 0$$.

Keywords: Laplacian, p-Laplacian,isoperimetric problem, shape optimisation, Robin boundary conditions, Wentzell boundary conditions.

: Primary 35P15; secondary (35J25, 35J60).

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 Wednesday, May 27, 2009