A boundary value problem for a PDE model in mass transfer theory: Representation of solutions and applications

Abstract

The system of partial differential equations $$ \left\{ \begin{array}{l}-{\rm div}(vDu)=f\quad {\rm in}\;{\rm \Omega}\\ |Du|-1=0\quad {\rm in }\;\{v > 0 \} \end{array} \right. $$ arises in the analysis of mathematical models for sandpile growth and in the context of the Monge–Kantorovich optimal mass transport theory. A representation formula for the solutions of a related boundary value problem is here obtained, extending the previous two-dimensional result of the first two authors to arbitrary space dimension. An application to the minimization of integral functionals of the form $$\int_{{\rm \Omega}} [h(|Du|)-f(x) u]{\rm d}x, $$ with f≥ 0, and h≥ 0 possibly non-convex, is also included.