Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Local Continuous Solvability of Equations Associated to Elliptic and
  Canceling Linear Differential Operators   Consider $A(x,D):C^{\infty}(\Omega,E) \rightarrow C^\infty(\Omega,F)$ an
elliptic and canceling linear differential operator of order $\nu$ with smooth
complex coefficients in $\Omega \subset \mathbb{R}^{N}$ from a finite dimension
complex vector space $E$ to a finite dimension complex vector space $F$ and
$A^{*}(x,D)$ {its} adjoint. In this work we characterize the (local) continuous
solvability of the partial differential equation $A^{*}(x,D)v=f$ (in the
distribution sense) for a given distribution $f$; more precisely we show that
any $x_0\in\Omega$ is contained in a neighborhood $U\subset \Omega$ in which
its continuous solvability is characterized by the following condition on $f$:
for every $\epsilon>0$ and any compact set $K \subset \subset U$, there exists
$\theta=\theta(K,\epsilon)>0$ such that the following holds for all smooth
function $\varphi$ supported in $K$: \begin{equation}\nonumber \left|
f(\varphi) \right| \leq \theta\|\varphi\|_{W^{\nu-1,1}} + \epsilon\|A(x,D)
\varphi\|_{L^{1}}, \end{equation} where $W^{\nu-1,1}$ stands for the homogenous
Sobolev space of all $L^1$ functions whose derivatives of order $\nu-1$ belongs
to $L^{1}(U)$. This characterization implies and extends results obtained
before for operators associated to elliptic complex of vector fields (see
\cite{MP}); we also provide local analogues, for a large range of differential
operators, to global results obtained for the classical divergence operator in
[4] and [9].

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The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
18
The categories of this paper are: econ
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: patt-sol
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The categories of this paper are: q-bio
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The categories of this paper are: cond-mat