01148nas a2200121   4500008004300000245006300043210006200106520076200168100002000930700002200950700001800972856003600990       2008                        en_Ud  00aTransition layer for the heterogeneous Allen-Cahn equation0 aTransition layer for the heterogeneous AllenCahn equation3 aWe 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 $-1<a(x)<1$ in $\\\\ov{\\\\Omega}$. We set $K$, $\\\\Omega_+$ and $\\\\Omega_-$ to be respectively the zero-level set of $a$, {a>0} 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 $j \\\\to + \\\\infty$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea1 aWei, Juncheng  uhttp://hdl.handle.net/1963/265600919nas a2200109   4500008004300000245008500043210006900128520053400197100002000731700002200751856003600773       2007                        en_Ud  00aConcentration on minimal submanifolds for a singularly perturbed Neumann problem0 aConcentration on minimal submanifolds for a singularly perturbed3 aWe 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 $1<p <\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/201301499nas a2200121   4500008004300000245014300043210006900186520102000255100002001275700002201295700002401317856003601341       2007                        en_Ud  00aSolutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part I: study of the limit set and approximate solutions0 aSolutions to the nonlinear Schroedinger equation carrying moment3 aWe prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide the characterization of the limit set, with natural stationarity and non-degeneracy conditions. We then construct an approximate solution up to order $\\\\epsilon^2$, showing that these conditions appear naturally in a Taylor expansion of the equation in powers of $\\\\epsilon$. Based on these, an existence result will be proved in the second part.1 aMahmoudi, Fethi1 aMalchiodi, Andrea1 aMontenegro, Marcelo  uhttp://hdl.handle.net/1963/211201231nas a2200109   4500008004300000245012500043210006900168520080600237100002001043700002201063856003601085       2007                        en_Ud  00aSolutions to the nonlinear Schroedinger equation carrying momentum along a curve. Part II: proof of the existence result0 aSolutions to the nonlinear Schroedinger equation carrying moment3 aWe prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- \\\\epsilon^2 \\\\Delta \\\\psi + V(x) \\\\psi = |\\\\psi|^{p-1} \\\\psi$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $\\\\epsilon$ a small parameter corresponding to the Planck constant. As $\\\\epsilon$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this work we identified the limit set and constructed approximate solutions, while here we give the complete proof of our main existence result.1 aMahmoudi, Fethi1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/211100921nas a2200109   4500008004300000245009700043210006900140520052400209100002000733700002200753856003600775       2006                        en_Ud  00aConcentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem0 aConcentration at manifolds of arbitrary dimension for a singular3 aWe consider the equation $- \\\\e^2 \\\\D u + u = u^p$ in $\\\\O \\\\subseteq \\\\R^N$, where $\\\\O$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of $\\\\pa \\\\O$, for $N \\\\geq 3$ and for $k \\\\in \\\\{1, \\\\dots, N-2\\\\}$. We impose Neumann boundary conditions, assuming $1<\\\\frac{N-k+2}{N-k-2}$ and $\\\\e \\\\to 0^+$. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.1 aMahmoudi, Fethi1 aMalchiodi, Andrea  uhttp://hdl.handle.net/1963/2170