Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Factorization for entropy production of the Eikonal equation and
  regularity   The Eikonal equation arises naturally in the limit of the second order
Aviles-Giga functional whose $\Gamma$-convergence is a long standing
challenging problem. The theory of entropy solutions of the Eikonal equation
plays a central role in the variational analysis of this problem. Establishing
fine structures of entropy solutions of the Eikonal equation, e.g.
concentration of entropy measures on $\mathcal{H}^1$-rectifiable sets in $2$D,
is arguably the key missing part for a proof of the full $\Gamma$-convergence
of the Aviles-Giga functional. In the first part of this work, for $p\in
\left(1,\frac{4}{3}\right]$ we establish an $L^p$ version of the main theorem
of Ghiraldin and Lamy [Comm. Pure Appl. Math. 73 (2020), no. 2, 317-349].
Specifically we show that if $m$ is a solution to the Eikonal equation, then
$m\in B^{\frac{1}{3}}_{3p,\infty,loc}$ is equivalent to all entropy productions
of $m$ being in $L^p_{loc}$. This result also shows that as a consequence of a
weak form of the Aviles-Giga conjecture (namely the conjecture that all
solutions to the Eikonal equation whose entropy productions are in $L^p_{loc}$
are rigid) - the rigidity/flexibility threshold of the Eikonal equation is
exactly the space $ B^{\frac{1}{3}}_{3,\infty,loc}$. In the second part of this
paper, under the assumption that all entropy productions are in $L^p_{loc}$, we
establish a factorization formula for entropy productions of solutions of the
Eikonal equation in terms of the two Jin-Kohn entropies. A consequence of this
formula is control of all entropy productions by the Jin-Kohn entropies in the
$L^p$ setting - this is a strong extension of an earlier result of the authors
[Annales de l'Institut Henri Poincar\'{e}. Analyse Non Lin\'{e}aire 35 (2018),
no. 2, 481-516].

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The categories of this paper are: stat
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: atom-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat