An extremal eigenvalue problem for the Wentzell-Laplace operator. (English) Zbl 1347.35186

Let \(d\geq2\) and let \(\Omega\) be a bounded connected open subset of \(\mathbb R^d\) with boundary of class \(C^{3}\) such that \(\int_{\partial\Omega}x=0\) and let \(\Delta_{\tau}\) be the Laplace-Beltrami operator on \(\partial\Omega\), \(\beta \in [0,+\infty[\), \((\lambda_{i,\beta}(\Omega))_{i \in\mathbb N}\) the increasing sequence of eigenvalues of the problem: \(-\Delta u=0\) in \(\Omega\), \(-\beta \Delta_{\tau}u + \partial_{n}u=\lambda u\) on \(\partial\Omega\), where \(n\) denotes the outward unit normal vector to \(\partial\Omega\). Let \(\Lambda[\Omega]\) be the spectral radius of the symmetric and positive semidefinite matrix \(P(\Omega)=(\int_{\partial\Omega} (\delta_{i,j} - n_{i}n_{j}))_{i,j=1,\dots,d}\).
Theorem. \[ \sum_{i=1}^{d} {1 \over \lambda_{i,\beta}(\Omega)} \geq {\int_{\partial\Omega}|x|^{2} \over |\Omega| + \beta \Lambda[\Omega]} \geq {d |B_{1}|^{-1/d}|\Omega|^{1+1/d} \over |\Omega|+ \beta \Lambda[\Omega]}\Big(1+(1+1/d)(2^{-2+1/d}-1/4)\big({|\Omega \Delta B| \over |B|}\big)^2\Big), \] where \(B_{1}\) is the unit ball and \(B\) is the ball of volume \(|\Omega|\) centered in \(0\); equality holds if \(\Omega\) is a ball.
The authors conjecture that the ball maximizes the first non trivial eigenvalue among smooth open sets of given volume and which are homeomorphic to the ball: to support this, they prove that balls are critical domains in any dimension and that they are local maximizers in dimension \(2\) and \(3\).


35P15 Estimates of eigenvalues in context of PDEs
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