Abstract. We prove an upper bound for the Aviles-Giga problem, which involves the minimization of the energy E ε (v) = ε ∇ 2 v 2 dx + ε −1 (1 − |∇v| 2 ) 2 dx over v ∈ H 2 ( ), where ε > 0 is a small parameter. Given v ∈ W 1,∞ ( ) such that ∇v ∈ BV and |∇v| = 1 a.e., we construct a family {v ε } satisfying: IntroductionConsider the energy functionalwhere is a C 2 bounded domain in R N , v is a scalar function and ε is a small parameter. Energies similar to (1.1) appear in different physical situations: smectic liquid crystals, blisters in thin films, micromagnetics (see [13] and the references therein). Clearly, one expects that any limit of the minimizers to (1.1) should satisfy the eikonal equation where J ∇v is the jump set of ∇v and ∇ ± v are the traces of ∇v on the two sides of the jump set (see Section 2 below for the exact definitions of the notions needed from the theory of functions of bounded variation). Most of the results on this problem treat the two-dimensional case N = 2 (an example due to De Lellis [6] shows that the AvilesGiga ansatz does not hold for N ≥ 3), so we assume N = 2 in the review of the known results below. Support for the Aviles-Giga conjecture was given in the work of Jin and A. Poliakovsky:

We study optimal interfacial structures in multiferroic materials with a biquadratic coupling between two order parameters. We discover a new duality relation between the strong coupling and the weak coupling regime for the case of isotropic gradient terms. We analyze the phase diagram depending on the coupling constant and anisotropy of the gradient term, and show that in a certain regime the secondary order parameter becomes activated only in the interfacial region.

Motivated by the construction of selfgravitating strings (cf Yang, 2001, 1994 [22,23]), we analyze a Liouville-type equation on the plane, derived in Yang (1994) [23]. We establish sharp existence and uniqueness properties for the corresponding radial solutions. We investigate also when the problem allows for non-radial solutions. (C) 2011 Elsevier Inc. All rights reserved

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Abstract. We study existence and uniqueness of positive eigenfunctions for the singular eigenvalue problem:|x| p on a bounded smooth domain Ω ⊂ R N with zero boundary condition. We also characterize all positive solutions of −∆ p u = |