Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotic behaviour of cuboids optimising Laplacian eigenvalues   We prove that in dimension $n \geq 2$, within the collection of unit measure
cuboids in $\mathbb{R}^n$ (i.e. domains of the form $\prod_{i=1}^{n}(0, a_n)$),
any sequence of minimising domains $R_k^\mathcal{D}$ for the Dirichlet
eigenvalues $\lambda_k$ converges to the unit cube as $k \to \infty$.
Correspondingly we also prove that any sequence of maximising domains
$R_k^\mathcal{N}$ for the Neumann eigenvalues $\mu_k$ within the same
collection of domains converges to the unit cube as $k\to \infty$. For $n=2$
this result was obtained by Antunes and Freitas in the case of Dirichlet
eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues.
The Dirichlet case for $n=3$ was recently treated by van den Berg and Gittins.
  In addition we obtain stability results for the optimal eigenvalues as $k \to
\infty$. We also obtain corresponding shape optimisation results for the Riesz
means of eigenvalues in the same collection of cuboids. For the Dirichlet case
this allows us to address the shape optimisation of the average of the first
$k$ eigenvalues.

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The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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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: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: stat
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom