Structure of entropy solutions to the eikonal equation

Abstract

In this paper, we establish rectifiability of the jump set of an ${\mathbf{S}}^1$–valued conservation law in two space–dimensions. This conservation law is a reformulation of the eikonal equation and is motivated by the singular limit of a class of variational problems. The only assumption on the weak solutions is that the entropy productions are (signed) Radon measures, an assumption which is justified by the variational origin. The methods are a combination of Geometric Measure Theory and elementary geometric arguments used to classify blow–ups.

The merit of our approach is that we obtain the structure as if the solutions were in BV, without using the BV–control, which is not available in these variationally motivated problems.