Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some regularity properties of viscosity solution defined by Hopf formula   Some properties of characteristic curves in connection with viscosity
solution of Hamilton-Jacobi equations $(H,\sigma)$ defined by Hopf formula
$u(t,x)=\max_{q\in\R^n}\{ \langle x,q\rangle -\sigma^*(q)-tH(q)\}$ are studied.
We are concerned with the points where the solution $u(t,x)$ is differentiable,
and the strip of the form $\mathcal R=(0,t_0)\times \R^n$ of the domain
$\Omega$ where $u(t,x)$ is of class $C^1(\mathcal R).$ Moreover, we investigate
the propagation of singularities in forward of this solution.

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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn