@article {2008,
title = {Transition layer for the heterogeneous Allen-Cahn equation},
journal = {Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631},
number = {arXiv.org;math/0702878v1},
year = {2008},
abstract = {We consider the equation $\\\\e^{2}\\\\Delta u=(u-a(x))(u^2-1)$ in $\\\\Omega$, $\\\\frac{\\\\partial u}{\\\\partial \\\\nu} =0$ on $\\\\partial \\\\Omega$, where $\\\\Omega$ is a smooth and bounded domain in $\\\\R^n$, $\\\\nu$ the outer unit normal to $\\\\pa\\\\Omega$, and $a$ a smooth function satisfying $-10} and {a<0}. Assuming $\\\\nabla a \\\\neq 0$ on $K$ and $a\\\\ne 0$ on $\\\\partial \\\\Omega$, we show that there exists a sequence $\\\\e_j \\\\to 0$ such that the above equation has a solution $u_{\\\\e_j}$ which converges uniformly to $\\\\pm 1$ on the compact sets of $\\\\O_{\\\\pm}$ as $j \\\\to + \\\\infty$.},
doi = {10.1016/j.anihpc.2007.03.008},
url = {http://hdl.handle.net/1963/2656},
author = {Fethi Mahmoudi and Andrea Malchiodi and Juncheng Wei}
}
@article {2007,
title = {Concentration on minimal submanifolds for a singularly perturbed Neumann problem},
journal = {Adv. Math. 209 (2007) 460-525},
number = {arXiv.org;math/0611558},
year = {2007},
abstract = {We consider the equation $- \\\\e^2 \\\\D u + u= u^p$ in $\\\\Omega \\\\subseteq \\\\R^N$, where $\\\\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\partial \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, ..., N-2\\\\}$. We impose Neumann boundary conditions, assuming $1