TY - JOUR
T1 - Transition layer for the heterogeneous Allen-Cahn equation
JF - Ann. Inst. H. Poincare Anal. Non Lineaire 25 (2008) 609-631
Y1 - 2008
A1 - Fethi Mahmoudi
A1 - Andrea Malchiodi
A1 - Juncheng Wei
AB - 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 $j \\\\to + \\\\infty$.
UR - http://hdl.handle.net/1963/2656
U1 - 1467
U2 - Mathematics
U3 - Functional Analysis and Applications
ER -
TY - JOUR
T1 - Concentration on minimal submanifolds for a singularly perturbed Neumann problem
JF - Adv. Math. 209 (2007) 460-525
Y1 - 2007
A1 - Fethi Mahmoudi
A1 - Andrea Malchiodi
AB - 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