*Published Paper*

**Inserted:** 2 sep 2014

**Last Updated:** 24 may 2017

**Journal:** Comm. Partial Differential Equations

**Volume:** 40

**Pages:** 1543–1557

**Year:** 2015

**Links:**
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**Abstract:**

We prove that any $2$-dimensional solution $\psi\in W_{loc}^{1+\frac 1 3, 3}$ of the eikonal equation has locally Lipschitz gradient $\nabla \psi$ except at a locally finite number of vortices.