Regularity of the Eikonal equation with two vanishing entropies

Abstract

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be a bounded simply-connected domain. The Eikonal equation $\left|\nabla u\right|=1$ for a function $u:\mathrm{\Omega }\subset {\mathbb{R}}^2\to \mathbb{R}$ has very little regularity, examples with singularities of the gradient existing on a set of positive $H^1$ measure are trivial to construct. With the mild additional condition of two vanishing entropies we show ∇u is locally Lipschitz outside a locally finite set. Our condition is motivated by a well known problem in Calculus of Variations known as the Aviles–Giga problem. The two entropies we consider were introduced by Jin, Kohn [26], Ambrosio, DeLellis, Mantegazza [2] to study the Γ-limit of the Aviles–Giga functional. Formally if u satisfies the Eikonal equation and if

where ${\tilde{\mathrm{\Sigma }}}_{e_1{e}_2}$ and ${\tilde{\mathrm{\Sigma }}}_{{ϵ}_1{ϵ}_2}$ are the entropies introduced by Jin, Kohn [26], and Ambrosio, DeLellis, Mantegazza [2], then ∇u is locally Lipschitz continuous outside a locally finite set.

Condition (1) is motivated by the zero energy states of the Aviles–Giga functional. The zero energy states of the Aviles–Giga functional have been characterized by Jabin, Otto, Perthame [25]. Among other results they showed that if ${\lim }_{n\to \infty }{I}_{{ϵ}_n}(u_n)=0$ for some sequence $u_n\in W_0^{2,2}(\mathrm{\Omega })$ and $u={\lim }_{n\to \infty }{u}_n$ then ∇u is Lipschitz continuous outside a finite set. This is essentially a corollary to their theorem that if u is a solution to the Eikonal equation $\left|\nabla u\right|=1$ a.e. and if for every “entropy” Φ (in the sense of [18], Definition 1) function u satisfies $\nabla \cdot \left[\mathrm{\Phi }(\nabla u^{\perp })\right]=0$ distributionally in Ω then ∇u is locally Lipschitz continuous outside a locally finite set. In this paper we generalize this result in that we require only two entropies to vanish.

The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion $DF\in K$ where $K\subset M^{2\times 2}$ is a connected compact set of matrices without Rank-1 connections. Equivalently this differential inclusion can be written as a constrained non-linear Beltrami equation. The set K is also non-elliptic in the sense of Sverak [32]. By use of this transformation and by utilizing ideas from the work on regularity of solutions of the Eikonal equation in fractional Sobolev space by Ignat [23], DeLellis, Ignat [15] as well as methods of Sverak [32], regularity is established.