Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp gradient estimates on weighted manifolds with compact boundary   In this paper, we prove sharp gradient estimates for positive solutions to
the weighted heat equation on smooth metric measure spaces with compact
boundary. As an application, we prove Liouville theorems for ancient solutions
satisfying the Dirichlet boundary condition and some sharp growth restriction
near infinity. Our results can be regarded as a refinement of recent results
due to Kunikawa and Sakurai.

33
The categories of this paper are: plasm-ph
25
The categories of this paper are: patt-sol
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
5
The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
9
The categories of this paper are: nlin
14
The categories of this paper are: math-ph
22
The categories of this paper are: chao-dyn