Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On uniform estimates for Laplace equation in balls with small holes   In this paper, we consider the Dirichlet problem of the three-dimensional
Laplace equation in the unit ball with a shrinking hole. The problem typically
arises from homogenization problems in domains perforated with tiny holes. We
give an almost complete description concerning the uniform $W^{1,p}$ estimates:
for any $3/2<p<3$ there hold the uniform $W^{1,p}$ estimates; for any $1<p<3/2$
or $3<p<\infty $, there are counterexamples indicating that the uniform
$W^{1,p}$ estimates do not hold. The results can be generalized to higher
dimensions.

10
The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-fin
1
The categories of this paper are: cs
24
The categories of this paper are: dg-ga