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

Abstract

Given a bounded domain Ω \Omega in R 2 \mathbb {R}^2 with smooth boundary, the cut locus Σ ¯ \overline \Sigma is the closure of the set of nondifferentiability points of the distance d d from the boundary of Ω \Omega . The normal distance to the cut locus, τ ( x ) \tau (x) , is the map which measures the length of the line segment joining x x to the cut locus along the normal direction D d ( x ) Dd(x) , whenever x ∉ Σ ¯ x\notin \overline \Sigma . Recent results show that this map, restricted to boundary points, is Lipschitz continuous, as long as the boundary of Ω \Omega is of class C 2 , 1 C^{2,1} . Our main result is the global Hölder regularity of τ \tau in the case of a domain Ω \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 Ω \Omega . The above regularity result for τ \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.