The Neumann problem for the ∞ -Laplacian and the Monge–Kantorovich mass transfer problem

We consider the natural Neumann boundary condition for the ∞ -Laplacian. We study the limit as p → ∞ of solutions of − Δ p u p = 0 in a domain Ω with | D u p | p − 2 ∂ u p / ∂ ν = g on ∂ Ω . We obtain a natural minimization problem that is verified by a limit point of { u p } and a limit problem that is satisfied in the viscosity sense. It turns out that the limit variational problem is related to the Monge–Kantorovich mass transfer problems when the measures are supported on ∂ Ω .