Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gradient Estimates on Dirichlet Eigenfunctions   By methods of stochastic analysis on Riemannian manifolds, we derive explicit
constants $c\_1(D)$ and $c\_2(D)$ for a $d$-dimensional compact Riemannian
manifold $D$ with boundary such that $c\_1(D)\sqrt{\lambda}\|\phi\|\_\infty \le
\|\nabla \phi\|\_\infty\le c\_2(D)\sqrt{\lambda} \|\phi\|\_\infty$ holds for
any Dirichlet eigenfunction $\phi$ of $-\Delta$ with eigenvalue $\lambda$. In
particular, when $D$ is convex with nonnegative Ricci curvature, this estimate
holds for $c\_1(D)=\frac{1}{de}$ and
$c\_2(D)=\sqrt{e}\left(\frac{\sqrt{2}}{\sqrt{\pi}}+\frac{\sqrt{\pi}}{4\sqrt{2}}\right)$.
Corresponding two-sided gradient estimates for Neumann eigenfunctions are
derived in the second part of the paper.

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The categories of this paper are: plasm-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: chem-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys