Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Extrapolation of the Dirichlet problem for elliptic equations with
  complex coefficients   In this paper, we prove an extrapolation result for complex coefficient
divergence form operators that satisfy a strong ellipticity condition known as
$p$-{\it ellipticity}. Specifically, let $\Omega$ be a chord-arc domain in
$\mathbb R^n$ and the operator $\mathcal L =
\partial_{i}\left(A_{ij}(x)\partial_{j}\right) +B_{i}(x)\partial_{i} $ be
elliptic, with $|B_i(x)| \le K\delta(x)^{-1}$ for a small $K$. Let $p_0 =
\sup\{p>1: A \,\,\text{is}\,\, \text{$p$-elliptic}\}$.
  We establish that if the $L^q$ Dirichlet problem is solvable for $\mathcal L$
for some $1<q< \frac{p_0(n-1)}{(n-2)}$, then the $L^p$ Dirichlet problem is
solvable for all $p$ in the range $[q, \frac{p_0(n-1)}{(n-2)})$. In particular,
if the matrix $A$ is real, or $n=2$, the $L^p$ Dirichlet problem is solvable
for $p$ in the range $[q, \infty)$.

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The categories of this paper are: cond-mat
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg