Another regularizing property of the 2D eikonal equation

A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\subset {\mathbb{R}}^2\to {\mathbb{R}}^2$ such that $\mathrm{\mid }m\mathrm{\mid }=1$ a.e. and $\mathrm{d}\mathrm{i}\mathrm{v}m=0$ in ${\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1\mathrm{/}3,3}$, 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 $m\in B_{p,\mathrm{\infty }}^{1\mathrm{/}3}$ 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 $6$.