Concentration on minimal submanifolds for a singularly perturbed Neumann problem

Abstract

We consider the equation −ε 2 ∆uwhere Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N ≥ 3 and for k ∈ {1, . . . , N − 2}. We impose Neumann boundary conditions, assuming 1 < p < N −k+2 N −k−2 and ε → 0 + . This result settles in full generality a phenomenon previously considered only in the particular case N = 3 and k = 1.

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

“…Similar resonance has been observed the problem of building foliations of a neighborhood of a geodesic by CMC tubes considered in [17,22]. This has also been the case for (simple) concentration phenomena for various elliptic problems, see [8,16,19,20].…”

“…The Laplace-Beltrami and Jacobi operators. If (M,g) is an N -dimensional Riemannian manifold, the Laplace-Beltrami operator on M is defined in local coordinates by the formula 16) whereg ab denotes the inverse of the matrix (g ab ). Let K ⊂ M be an (N − 1)-dimensional closed smooth embedded submanifold associated with the metricg 0 induced from (M,g).…”

Resonance Phenomena in a Singular Perturbation Problem in the Case of Exchange of Stabilities

“…Such a phenomenon is not new and, in the context of singularly perturbed semilinear elliptic equations, was originally found by A. Malchiodi and M. Montenegro in [34]. Since this seminal paper, this phenomenon has also been found in other instances, for example in the study of other semilinear partial differential equations [15,16,31,33,36] or in the study of constant mean curvature surfaces [32,35]. Loosely speaking, it is caused by the presence of the tangential dimension θ along the curve Γ and the fact that the profile in the normal t direction in unstable (see the discussion in the next paragraph).…”

“…The desired result for the full problem (11.1) then follows directly from Schauder's fixed point theorem. In fact, refining the fixed-point region, we can actually get [15], which we use in the present paper, the preferred approach seems to be that of [34] (see [31,33,36]). Let us briefly discuss this point and what are the difficulties in employing the scheme of [34] in the present situation.…”

“…Our construction of the approximate solution can easily be adapted to problem (1.1) posed in N dimensions. Based on the radially symmetric case, we expect that the eigenvalues of the linearization of (1.1) around this approximation should behave qualitatively as Λ i described in (1.8) where τ i now are the eigenvalues of the Laplace-Beltrami operator of the manifold M. (Instead of expanding in polar coordinates, one can use (naively) a Fourier decomposition with respect to the eigenfunctions of the Laplace-Beltrami operator and the normal Laplacian on the manifold M, see [31] for a related result). Assuming that this expectation holds, by Weyl's asymptotic formula, the eigenvalues that are closest to zero should behave qualitatively like…”