Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Regularity of the extremal solution in a MEMS model with advection   We consider the regularity of the extremal solution of the nonlinear
eigenvalue problem (S)_\lambda \qquad {rcr} -\Delta u + c(x) \cdot \nabla u &=&
\frac{\lambda}{(1-u)^2} \qquad {in $ \Omega$}, u &=& 0 \qquad {on $ \pOm$},
where $ \Omega $ is a smooth bounded domain in $ \IR^N$ and $ c(x)$ is a smooth
bounded vector field on $\bar \Omega$. We show that, just like in the
advection-free model ($c\equiv 0$), all semi-stable solutions are smooth if
(and only if) the dimension $N\leq 7$. The novelty here comes from the lack of
a suitable variational characterization for the semi-stability assumption. We
overcome this difficulty by using a general version of Hardy's inequality. In a
forthcoming paper \cite{CG2}, we indicate how this method applies to many other
nonlinear eigenvalue problems involving advection (including the Gelfand
problem), showing that they all essentially have the same critical dimension as
their advection-free counterparts.

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The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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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: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: cs
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex