Regularity of the Eikonal equation with two vanishing entropies |
| |
Authors: | Andrew Lorent Guanying Peng |
| |
Institution: | Mathematics Department, University of Cincinnati, 2600 Clifton Ave., Cincinnati, OH 45221, United States |
| |
Abstract: | Let be a bounded simply-connected domain. The Eikonal equation for a function has very little regularity, examples with singularities of the gradient existing on a set of positive 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 (1) where and 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 for some sequence and 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 a.e. and if for every “entropy” Φ (in the sense of 18], Definition 1) function u satisfies 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 where 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. |
| |
Keywords: | 28A75 Eikonal equation Aviles Giga functional Entropies Non-linear Beltrami equation Differential inclusions |
本文献已被 ScienceDirect 等数据库收录! |
|