Upper bounds for singular perturbation problems involving gradient fields

Abstract

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:

Upper bounds for a class of energies containing a non-local term

“…5.1) we proved the following statement. This approximation result generalize Claim 3 of Lemma 3.4 from [6] and was an essential tool in the optimizing the upper bound in [8]. …”

“…of (3.37), over all kernels η ∈ V (d) analogously to what was done in [6,8]. By the same method as there, we can obtain the following result.…”

“…In this work, using the technique developed in [6][7][8], we construct the upper bound as ε ↓ 0 for the general energy of the form (1.1) under certain conditions on M for functions φ ∈ BV ∩ L ∞ . This is included in Theorems 3.1 and 3.2.…”

Entropy method for line-energies

“…The motivation comes either from solid mechanics, liquid crystals or micromagnetic models (see [7,14]). It gave rise to a series of articles [20,6,2,12,8,23] that justify that I f with f (t) = t 3 /6 (given by (4)) is indeed the asymptotic energy of {G ε }.…”

Lower Bound for the Energy of Bloch Walls in Micromagnetics

“…The difficulty consists in the upper bound construction for admissible configurations m 0 : recovery sequences have been constructed only for BV configurations m 0 (see Conti and De Lellis [8] and Poliakovsky [20]). We emphasize that the difference between the line-energy density associated to jumps of m 0 in E 0 and AG 0 comes from the two different anisotropy terms: …”