Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A variant of H\"ormander's $L^2$ existence theorem for Dirac operator in
  Clifford analysis   In this paper, we give the H\"ormander's $L^2$ theorem for Dirac operator
over an open subset $\Omega\in\R^{n+1}$ with Clifford algebra. Some sufficient
condition on the existence of the weak solutions for Dirac operator has been
found in the sense of Clifford analysis. In particular, if $\Omega$ is bounded,
then we prove that for any $f$ in $L^2$ space with value in Clifford algebra,
there exists a weak solution of Dirac operator such that $$\bar{D}u=f$$ with
$u$ in the $L^2$ space as well. The method is based on H\"ormander's $L^2$
existence theorem in complex analysis and the $L^2$ weighted space is utilised.

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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: econ
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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: stat
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph