Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains   In this paper we study the $L^p$ boundary value problems for
$\mathcal{L}(u)=0$ in $\mathbb{R}^{d+1}_+$, where
$\mathcal{L}=-\text{div}(A\nabla)$ is a second order elliptic operator with
real and symmetric coefficients. Assume that $A$ is {\it periodic} in $x_{d+1}$
and satisfies some minimal smoothness condition in the $x_{d+1}$ variable, we
show that the $L^p$ Neumann and regularity problems are uniquely solvable for
$1<p<2+\delta$. We also present a new proof of Dahlberg's theorem on the $L^p$
Dirichlet problem for $2-\delta<p< \infty$ (Dahlberg's original unpublished
proof is given in the Appendix). As the periodic and smoothness conditions are
imposed only on the $x_{d+1}$ variable, these results extend directly from
$\mathbb{R}^{d+1}_+$ to regions above Lipschitz graphs. Consequently, by
localization techniques, we obtain uniform $L^p$ estimates for the Dirichlet,
Neumann and regularity problems on bounded Lipschitz domains for a family of
second order elliptic operators arising in the theory of homogenization. The
results on the Neumann and regularity problems are new even for smooth domains.

12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con