An optimization problem with volume constraint with applications to optimal mass transport

Abstract

In this manuscript we study the following optimization problem with volume constraint: min{ [Formula presented] ∫Ω|∇v|pdx−∫∂ΩgvdHN−1:v∈W1,p(Ω), and LN({v>0})≤α}. Here Ω⊂RN is a bounded and smooth domain, g is a continuous function and α is a fixed constant such that 00 we prove that a minimizer exists and satisfies {−Δpup=0in {up>0}∪{up<0},|∇up|p−2 [Formula presented] =gon ∂Ω∩∂({up>0}∪{up<0}),LN({up>0})=α. Next, we analyze the limit as p→∞. We obtain that any sequence of weak solutions converges, up to a subsequence, limpj→∞⁡upj (x)=u∞(x), uniformly in Ω‾, and uniform limits, u∞, are solutions to the maximization problem with volume constraint max⁡{∫∂ΩgvdHN−1:v∈W1,∞(Ω),‖∇v‖L∞(Ω)≤1 and LN({v>0})≤α}. Furthermore, we obtain the limit equation that is verified by u∞ in the viscosity sense. Finally, it turns out that such a limit variational problem is connected to the Monge-Kantorovich mass transfer problem with the involved measures are supported on ∂Ω and along the limiting free boundary, ∂{u∞≠0}. Furthermore, we show some explicit examples of solutions for certain configurations of the domain and data.