Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Clustering of Boundary Interfaces for an inhomogeneous Allen-Cahn
  equation on a smooth bounded domain   We consider the inhomogeneous Allen-Cahn equation $$ \epsilon^2\Delta
u\,+\,V(y)(1-u^2)\,u\,=\,0\quad \mbox{in}\ \Omega, \qquad \frac {\partial
u}{\partial \nu}\,=\,0\quad \mbox{on}\ \partial \Omega, $$ where $\Omega$ is a
bounded domain in ${\mathbb R}^2$ with smooth boundary $\partial\Omega$ and
$V(x)$ is a positive smooth function, $\epsilon>0$ is a small parameter, $\nu$
denotes the unit outward normal of $\partial\Omega$. For any fixed integer
$N\geq 2$, we will show the existence of a clustered solution $u_{\epsilon}$
with $N$-transition layers near $\partial \Omega$ with mutual distance
$O(\epsilon|\ln \epsilon|)$, provided that the generalized mean curvature
$\mathcal{H} $ of $\partial\Omega$ is positive and $\epsilon$ stays away from a
discrete set of values at which resonance occurs. Our result is an extension of
those (with dimension two) by A. Malchiodi, W.-M. Ni, J. Wei in Pacific J.
Math. (Vol. 229, 2007, no. 2, 447-468) and A. Malchiodi, J. Wei in J. Fixed
Point Theory Appl. (Vol. 1, 2007, no. 2, 305-336)

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The categories of this paper are: comp-gas
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an