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

In this manuscript we study the following optimization problem with volume constraint: min {1/p integral(Omega) vertical bar del v vertical bar(P) dx - integral(partial derivative Omega) g v d HN-1: v is an element of W-1,W-p (Omega), and L-N({v >0}) >= alpha}. Here Omega subset of R-N is a bounded and smooth domain, alpha is a continuous function and a is a fixed constant such that 0 < alpha < L-N (Omega). Under the assumption that integral(partial derivative Omega) g(x)d HN-1 > 0 we prove that a minimizer exists and a satisfies {-Delta(p)u(p) = 0 in {u(p) > 0} boolean OR {u(p) < 0}, vertical bar del u(p)vertical bar(p-2) partial derivative u(p)/partial derivative eta = g on partial derivative Omega boolean AND partial derivative({u(p) > 0} boolean OR {u(p) <0}), L-N ({u(p) >0}) = alpha. Next, we analyze the limit as p -> infinity. We obtain that any sequence of weak solutions converges, up to a subsequence, lim(pj ->infinity) u(pj) (x)= u(infinity) (x), uniformly in (Omega) over bar, and uniform limits, u(infinity), are solutions to the maximization problem with volume constraint max {integral(partial derivative Omega) g v d HN-1: v is an element of W-1,W-infinity (Omega), parallel to del v parallel to(L infinity(Omega)) <= 1 and L-N ({v > 0) <= alpha}. Furthermore, we obtain the limit equation that is verified by u(infinity) 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 partial derivative Omega and along the limiting free boundary, partial derivative{u(infinity)not equal 0}. Furthermore, we show some explicit examples of solutions for certain configurations of the domain and data. (C) 2019 Elsevier Inc. All rights reserved.