The optimal Kobayashi-distance bounds previously known for C^{2,alpha}-smooth strongly pseudoconvex domains are proved for general C^2 strongly pseudoconvex domains, and upper bounds are extended to C^{1,1} and to non-semipositive points.
Boundary regularity for the distance functions, and the eikonal equation
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abstract
We study the gain in regularity of the distance to the boundary of a domain in $\mathbb R^m$. 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 $\mathcal C^{1,1}$ 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.
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Optimal bounds for the Kobayashi distance near $\mathcal C^2$-smooth boundary points
The optimal Kobayashi-distance bounds previously known for C^{2,alpha}-smooth strongly pseudoconvex domains are proved for general C^2 strongly pseudoconvex domains, and upper bounds are extended to C^{1,1} and to non-semipositive points.