ABSTRACT. We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain Ω Ă R n , n ě 3, we prove that if the mean curvature H of the boundary obeys the conditiońthen Ω is a round ball. MSC (2010): 35N25, 31B15, 35B06, 53C21.

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