Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp solvability criteria for Dirichlet problems of mean curvature type
  in Riemannian manifolds: non-existence results   It is well known that the Serrin condition is a necessary condition for the
solvability of the Dirichlet problem for the prescribed mean curvature equation
in bounded domains of $\mathbb{R}^n$ with certain regularity. In this paper we
investigate the sharpness of the Serrin condition for the vertical mean
curvature equation in the product $ M^n \times \mathbb{R} $. Precisely, given a
$\mathscr{C}^2$ bounded domain $\Omega$ in $M$ and a function $ H = H (x, z) $
continuous in $\overline{\Omega}\times\mathbb{R}$ and non-decreasing in the
variable $z$, we prove that the strong Serrin condition
$(n-1)\mathcal{H}_{\partial\Omega}(y)\geq n\sup\limits_{z\in\mathbb{R}}|H(y,z)|
\ \forall \ y\in\partial\Omega $, is a necessary condition for the solvability
of the Dirichlet problem in a large class of Riemannian manifolds within which
are the Hadamard manifolds and manifolds whose sectional curvatures are bounded
above by a positive constant. As a consequence of our results we deduce
Jenkins-Serrin and Serrin type sharp solvability criteria.

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The categories of this paper are: hep-th
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
34
The categories of this paper are: chem-ph
32
The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-lat