ON THE ISOPERIMETRIC PROBLEM FOR THE LAPLACIAN WITH ROBIN AND WENTZELL BOUNDARY CONDITIONS
James Bernard Kennedy · Bulletin of the Australian Mathematical Society · 2010
We consider the problem of minimising the eigenvalues of the Laplacian with Robin boundary conditions $\\frac{\\partial u}{\\partial \ u} + \\alpha u = 0$ and generalised Wentzell boundary conditions $\\Delta u + \\beta \\frac{\\partial u}{\\partial \ u} + \\gamma u = 0$ with respect to the domain $\\Omega \\subset \\mathbb R^N$ on which the problem is defined. For the Robin problem, when $\\alpha > 0$ we extend the Faber-Krahn inequality of Daners [Math. Ann. 335 (2006), 767--785], which states that the ball minimises the first eigenvalue, to prove that the minimiser is unique amongst domains of class $C^2$. The method of proof uses a functional of the level sets to estimate the first eigenvalue from below, together with a rearrangement of the ball's eigenfunction onto the domain $\\Omega$ and the usual isoperimetric inequality. We then prove that the second eigenvalue attains its minimum only on the disjoint union of two equal balls, and set the proof up so it works for the Robin $p$-Laplacian. For the higher eigenvalues, we show that it is in general impossible for a minimiser to exist independently of $\\alpha > 0$. When $\\alpha 0$ establish a type of equivalence property between the Wentzell and Robin minimisers for all eigenvalues. This yields a minimiser of the second Wentzell eigenvalue. We also prove a Cheeger-type inequality for the first eigenvalue in this case.