Hölder regularity of the normal distance with an application to a PDE model for growing sandpiles

Abstract:

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.