Fethi Mahmoudi

Transition layer for the heterogeneous Allen-Cahn equation

Abstract We consider the equation (1) e 2 Δ u = ( u − a ( x ) ) ( u 2 − 1 ) in Ω , ∂ u ∂ ν = 0 on ∂ Ω , where Ω is a smooth and bounded domain in R n , ν the outer unit normal to ∂Ω, and a a smooth function satisfying − 1 a ( x ) 1 in Ω ¯ . We set K, Ω + and Ω − to be respectively the zero-level set of a, { a > 0 } and { a 0 } . Assuming ∇ a ≠ 0 on K and a ≠ 0 on ∂Ω, we show that there exists a sequence e j → 0 such that Eq. (1) has a solution u e j which converges uniformly to ±1 on the compact sets of Ω ± as j → + ∞ . This result settles in general dimension a conjecture posed in [P. Fife, M.W. Greenlee, Interior transition layers of elliptic boundary value problem with a small parameter, Russian Math. Surveys 29 (4) (1974) 103–131], proved in [M. del Pino, M. Kowalczyk, J. Wei, Resonance and interior layers in an inhomogeneous phase transition model, SIAM J. Math. Anal. 38 (5) (2007) 1542–1564] only for n = 2 .

Bubbling on boundary submanifolds for the Lin–Ni–Takagi problem at higher critical exponents

We consider the equation

Solutions to the nonlinear Schrödinger equation carrying momentum along a curve

d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega

Nonlinear elliptic problem related to the Hardy inequality with singular term at the boundary

, under zero Neumann boundary conditions, where

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

\Omega

Ground states and concentration phenomena for the fractional Schrödinger equation

is open, smooth and bounded and

Constant mean curvature hypersurfaces condensing on a submanifold

d

On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

is a small positive parameter. We assume that there is a

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

k

Concentration at manifolds of arbitrary dimension for a singularly perturbed Neumann problem

-dimensional closed, embedded minimal submanifold