Résumé (EN)

Given a bounded domain $ \Omega$ in $ \mathbb{R}^2$ with smooth boundary, the cut locus $ \overline \Sigma$ is the closure of the set of nondifferentiability points of the distance $ d$ from the boundary of $ \Omega$. The normal distance to the cut locus, $ \tau(x)$, is the map which measures the length of the line segment joining $ x$ to the cut locus along the normal direction $ Dd(x)$, whenever $ x\notin \overline \Sigma$. Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of $ \Omega$ is of class $ C^{2,1}$. Our main result is the global Hölder regularity of $ \tau$ in the case of a domain $ \Omega$ with analytic boundary. We will also show that the regularity obtained is optimal, as soon as the set of the so-called regular conjugate points is nonempty. In all the other cases, Lipschitz continuity can be extended to the whole domain $ \Omega$. The above regularity result for $ \tau$ is also applied to derive the Hölder continuity of the solution of a system of partial differential equations that arises in granular matter theory and optimal mass transfer.