Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity theory for solutions to second order elliptic operators with
  complex coefficients and the $L^p$ Dirichlet problem   We establish a new theory of regularity for elliptic complex valued second
order equations of the form $\mathcal L=$div$A(\nabla\cdot)$, when the
coefficients of the matrix $A$ satisfy a natural algebraic condition, a
strengthened version of a condition known in the literature as
$L^p$-dissipativity. Precisely, the regularity result is a reverse H\"older
condition for $L^p$ averages of solutions on interior balls, and serves as a
replacement for the De Giorgi - Nash - Moser regularity of solutions to
real-valued divergence form elliptic operators. In a series of papers, Cialdea
and Maz'ya studied necessary and sufficient conditions for $L^p$-dissipativity
of second order complex coefficient operators and systems. Recently, Carbonaro
and Dragi\v{c}evi\'c introduced a condition they termed $p$-ellipticity, and
showed that it had implications for boundedness of certain bilinear operators
that arise from complex valued second order differential operators. Their
$p$-ellipticity condition is exactly our strengthened version of
$L^p$-dissipativity. The regularity results of the present paper are applied to
solve $L^p$ Dirichlet problems for $\mathcal
L=$div$A(\nabla\cdot)+B\cdot\nabla$ when $A$ and $B$ satisfy a natural and
familiar Carleson measure condition. We show solvability of the $L^p$ Dirichlet
boundary value problem for $p$ in the range where $A$ is $p$-elliptic.

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The categories of this paper are: astro-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: nlin
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The categories of this paper are: atom-ph
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-th
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The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nucl-ex
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The categories of this paper are: funct-an
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The categories of this paper are: hep-th
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: econ