Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Mean curvature bounds and eigenvalues of Robin Laplacians   We consider the Laplacian with attractive Robin boundary conditions, \[
Q^\Omega_\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u
\text{ on } \partial\Omega, \] in a class of bounded smooth domains
$\Omega\in\mathbb{R}^\nu$; here $n$ is the outward unit normal and $\alpha>0$
is a constant. We show that for each $j\in\mathbb{N}$ and $\alpha\to+\infty$,
the $j$th eigenvalue $E_j(Q^\Omega_\alpha)$ has the asymptotics \[
E_j(Q^\Omega_\alpha)=-\alpha^2 -(\nu-1)H_\mathrm{max}(\Omega)\,\alpha+{\mathcal
O}(\alpha^{2/3}), \] where $H_\mathrm{max}(\Omega)$ is the maximum mean
curvature at $\partial \Omega$. The discussion of the reverse Faber-Krahn
inequality gives rise to a new geometric problem concerning the minimization of
$H_\mathrm{max}$. In particular, we show that the ball is the strict minimizer
of $H_\mathrm{max}$ among the smooth star-shaped domains of a given volume,
which leads to the following result: if $B$ is a ball and $\Omega$ is any other
star-shaped smooth domain of the same volume, then for any fixed
$j\in\mathbb{N}$ we have $E_j(Q^B_\alpha)>E_j(Q^\Omega_\alpha)$ for large
$\alpha$. An open question concerning a larger class of domains is formulated.

31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: cs
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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
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The categories of this paper are: stat