Another regularizing property of the 2D eikonal equation

Abstract

A weak solution of the two-dimensional eikonal equation amounts to a vector field m:Ω⊂R2→R2 such that |m|=1 a.e. and divm=0 in D′(Ω). It is known that, if m has some low regularity, e.g., continuous or W1/3,3, then m is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption m∈Bp,∞1/3 for p>3. In this note we establish that regularizing effect in the borderline case p=6, above which the Besov regularity assumption implies continuity. If the domain is a disk and m satisfies tangent boundary conditions, we also prove this for p slightly below 6.