Free boundary regularity in the optimal partial transport problem

In the optimal partial transport problem, one is asked to transport a fraction 0 < m <= min {parallel to f parallel to(L1), parallel to g parallel to(L1)) of the mass of f = f chi(Omega) onto g = g chi(Lambda) while minimizing a transportation cost. If f and g are bounded away from zero and infinity on strictly convex domains Omega and Lambda, respectively, and if the cost is quadratic, then away from partial derivative (Omega boolean AND Lambda) the free boundaries of the active regions are shown to be C-loc(l,alpha) hypersurfaces up to a possible singular set. This improves and generalizes a result of Caffarelli and McCann (2010) [6] and solves a problem discussed by Figalli (2010) [8, Remark 4.15]. Moreover, a method is developed to estimate the Hausdorff dimension of the singular set: assuming Omega and Lambda to be uniformly convex domains with C-1,C-1 boundaries, we prove that the singular set is Hn-2 sigma-finite in the general case and Hn-2 finite if Omega and Lambda are separated by a hyperplane. (C) 2013 Elsevier Inc. All rights reserved.