Multidimensional boundary layers for a singularly perturbed Neumann problem

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains

“…A conjecture, see [N], states that generically one should find spike-layers of any dimension k running from 1 to n − 1. The present results, combined with those of [MM1] and [MM2], proves the validity of this conjecture in dimensions 2 and 3.…”

“…In this procedure, we also use Fourier cancellations taking advantage of the different frequencies of the resonant eigenvalues µ l,ε and σ l,ε . We notice that, in the present case, the characterization of the eigenfunctions of I ε (u k,ε ) is more difficult than in [MM1] or [MM2], because of the presence of the transversal eigenvalues.…”

“…The proof follows that of Proposition 2.3 in [MM2], and hence we omit it here. It is still based on Theorem 2.3 and some elementary inequalities.…”

“…As a first step, as in [MM2], we find approximate solutions to (P ε ) by iterative approximations. Using the coordinates y defined above, we look for a solution of the form u k,ε (y) = [w 0 + εw 1 + · · ·+ ε k w k ] εy 1 , y 2 + f 0 (εy 1 )+· · ·+ ε k−2 f k−2 (εy 1 ), y 3 .…”

On De Giorgi's conjecture in dimension N≥9

“…In the noncompact setting, for both (1.1) and nonlinear Schrödinger equation, solutions have been constructed by del Pino, Kowalczyk and Wei [12], del Pino, Kowalczyk, Pacard and Wei [11], [10], and Malchiodi [24]. See also Malchiodi and Montenegro [25], [26]. We should emphasize here the importance of our earlier works [11], [10] in the context of the present paper, and especially the idea of constructing solutions concentrating on a family of unbounded sets, all coming from a suitably rescaled basic set.…”