Boundary regularity for the distance functions, and the eikonal equation

We study the gain in regularity of the distance to the boundary of a domain in ᵐ. In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be C1,1 regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz.