Multidimensional boundary layers for a singularly perturbed Neumann problem

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Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature (2010, methodology): “. The result will be a consequence of various considerations on the asymptotic behavior of the small eigenvalues of Lσ as σ→ 0. The general scheme below has already been used in related settings, see [19, 20, 21, 16, 17], using the theory of smooth and analytic dependence of eigenvalues of families ”
MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2 (2010, background): “For all ε > 0 sufficiently small, there exists an entire solution uε of the Allen-Cahn equation (1.1) whose nodal set is the union of k disjoint curves”
Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains (2005, methodology): “One of the main aspects of the results in [28] and [ 29 ] was the presence of eigenvalues (of the linearized operator) which turn from positive into negative when " tends to zero.”
Some new entire solutions of semilinear elliptic equations on Rn (2009, background): “For problem (Pε), in [43], [44] it has been proved that, given any n ∈ N and any p > 1, there exist solutions concentrating at the whole boundary of Ω if ε tends to zero along a suitable sequence εj , corresponding to k = n − 1.”
The Toda System and Clustering Interfaces in the Allen–Cahn equation (2008, background): “Single-higher dimensional concentration phenomena has been the subject in, among other works, [12,25,26].”