Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and structure of minimizers of least gradient problems   We study existence of minimizers of the general least gradient problem
\[\inf_{u \in BV_f} \int_{\Omega}\varphi(x,Du),\] where $BV_f=\{u \in
BV(\Omega): \ \ u|_{\partial \Omega}=f\}$, $f\in L^{1}(\partial \Omega)$, and
$\varphi(x,\xi)$ is convex, continuous, and homogeneous function of degree $1$
with respect to the $\xi$ variable. It is proven that there exists a divergence
free vector field $T\in (L^{\infty}(\Omega))^n$ that determines the structure
of level sets of all (possible) minimizers, i.e. $T$ determines
$\frac{Du}{|Du|}$, $|Du|-$ a.e. in $\Omega$, for all minimizers $u$. We also
prove that every minimizer of the above least gradient problem is also a
minimizer of \[\inf_{u\in \mathcal{A}_f} \int_{\R^n}\varphi(x,Du),\] where
$\mathcal{A}_f=\{v\in BV(\R^n): \ \ v=f \ \ \hbox{on}\ \ \Omega^c\}$ and $f\in
W^{1,1}(\R^n)$ is a compactly supported extension of $f\in L^1(\partial
\Omega)$, and show that $T$ also determines the structure of level sets of all
minimizers of the latter problem. This relationship between minimizers of the
above two least gradient problems could be exploited to obtain information
about existence and structure of minimizers of the former problem from that of
the latter, which always exist.

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The categories of this paper are: mtrl-th
6
The categories of this paper are: astro-ph
9
The categories of this paper are: nlin
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: plasm-ph