Regularity of the Eikonal equation with two vanishing entropies

The method of proof is to transform any solution of the Eikonal equation satisfying (1) into a differential inclusion DF∈K where K⊂M2×2 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.