## 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 $$3$$.

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs
Full Text:

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