Characterization of Minimizers of Aviles–Giga Functionals in Special Domains

Abstract

We consider the singularly perturbed problem $$F_\varepsilon (u,\Omega ):=\int _\Omega \varepsilon |\nabla ^2u|^2 + \varepsilon ^{-1}|1-|\nabla u|^2|^2$$ F ε ( u , Ω ) : = ∫ Ω ε | ∇ 2 u | 2 + ε - 1 | 1 - | ∇ u | 2 | 2 on bounded domains $$\Omega \subset {\mathbb {R}}^2$$ Ω ⊂ R 2 . Under appropriate boundary conditions, we prove that if $$\Omega $$ Ω is an ellipse, then the minimizers of $$F_\varepsilon (\cdot ,\Omega )$$ F ε ( · , Ω ) converge to the viscosity solution of the eikonal equation $$|\nabla u|=1$$ | ∇ u | = 1 as $$\varepsilon \rightarrow 0$$ ε → 0 .

On a generalized Aviles-Giga functional: compactness, zero-energy states, regularity estimates and energy bounds

“…Therefore, Φ is an entropy for equation (24) in this subclass if and only if Φ = Φ • X −1 : ∂B → R 2 belongs to ENT, as follows directly from the definition (4) of the class ENT. Moreover we have Φ(m) = Φ(m).…”

“…For scalar conservation laws (3) with f uniformly convex (Burgers' equation), this rectifiability property has recently been proved in [23]. The results of [9] and [24,Proposition 1.7] can likely be generalized to the class of energy functionals (1) associated with any strictly convex C 1 norm • (using the kinetic formulation obtained in Lemma 19), but here we don't address that question and concentrate instead on optimal regularity estimates for solutions of the generalized Eikonal equation (2) whose entropy productions are locally finite Radon measures. In the classical case • = | • |, it is proved in [13] (adapting an argument of [14] for scalar conservation laws) that such solutions must locally have the Besov regularity…”

“…Solutions of ( 24) and of the generalized Eikonal equation ( 2) are in correspondence via the Lipschitz homeomorphism X defined in (12). Specifically, a map m solves (24) if and only if m = X(m) solves ( 2). This transformation also induces a correspondence between entropies.…”

“…This transformation also induces a correspondence between entropies. A C 1 map Φ : S 1 → R 2 is an entropy for equation (24) if and only if ∇ • Φ(m) = 0 for any smooth solution of (24). It is an exercise to see that this is equivalent to (d/dθ)Φ(e iθ ) being colinear to γ (θ).…”

“…The energy lower bound ensures that entropy productions are finite Radon measures, and a central question to solve the Γ-convergence conjecture for the classical Aviles-Giga functional is whether these measures are concentrated on a one-dimensional rectifiable set. Substantial progress on that question has been made in [9,24] but it remains open. For scalar conservation laws (3) with f uniformly convex (Burgers' equation), this rectifiability property has recently been proved in [23].…”