A boundary value problem for a PDE model in mass transfer theory: representations of solutions and regularity results

Given a bounded domain Ω ⊂ R n , let us denote by d(·) : Ω → R the distance function from the boundary ∂Ω. The set of points x ∈ Ω at which d is not differentiable is called the singular set of d and denoted by Σ. Its closure is often referred to as the cut locus. We introduce the map τ : Ω → R, defined by τ (x) = min  t ≥ 0 : x + tDd(x) ∈ Σ for all x ∈ Ω\Σ, τ (x) = 0 on Σ, which is sometimes called the maximal retraction length of Ω onto Σ or normal distance to Σ. The aim of this work is two–sided: 1 To present a global regularity result on the normal distance to the cut locus, showing that in the case when n = 2 and Ω is a bounded simply connected domain with analytic boundary, then τ is either a Lipschitz continuous or a H¨older continuous function of exponent at least 2/3. We apply this result to the study of regularity of the solutions (in a suitable sense) of system (1)    − div (vDu) = f in Ω |Du| − 1 = 0 in {v > 0} |Du| ≤ 1 u, v ≥ 0 in Ω u = 0 on ∂Ω. where f : Ω → R is a non–negative Lipschitz continuous function. It turns out that the second component v of the solution of (1) is either Lipschitz continuous or H¨older continuous, as well. 2 To show an existence/uniqueness result on the solutions of system (1) in any space dimension, generalizing what recently found in dimension n = 2 in the context of granular matter theory