Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extremal functions for Morrey's inequality in convex domains   For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's
inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le
\int_\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space
$W^{1,p}_0(\Omega)$. We show that the ratio of any two extremal functions is
constant provided that $\Omega$ is convex. We also explain why this property
fails to hold in general and verify that convexity is not a necessary condition
for a domain to have this property. As a by product, we obtain the uniqueness
of an optimization problem involving the Green's function for the
$p$-Laplacian.

37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
17
The categories of this paper are: q-fin
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph