Variational Approach to Regularity of Optimal Transport Maps: General Cost Functions

Abstract

We extend the variational approach to regularity for optimal transport maps initiated by Goldman and the first author to the case of general cost functions. Our main result is an $$\epsilon $$ ϵ -regularity result for optimal transport maps between Hölder continuous densities slightly more quantitative than the result by De Philippis–Figalli. One of the new contributions is the use of almost-minimality: if the cost is quantitatively close to the Euclidean cost function, a minimizer for the optimal transport problem with general cost is an almost-minimizer for the one with quadratic cost. This further highlights the connection between our variational approach and De Giorgi’s strategy for $$\epsilon $$ ϵ -regularity of minimal surfaces.

Geometric regularisation for optimal transport with strongly p-convex cost

“…The key tool in the proof was a version of Theorem 1.1 in this setting. In later papers, continuous densities [5], rougher measures [7], more general cost functions (albeit still close to the quadratic cost functional) [15], as well as almost-minimisers of the quadratic cost functional [15] were considered. The quadratic version of Theorem 1.1 was also used to provide a more refined linearisation result of (1.2) in the quadratic set-up in [7] and of a similar statement in the context of optimal matching in [6].…”

Geometric linearisation for optimal transport with strongly p-convex cost

“…The key tool in the proof was a version of Theorem 1.1 in this setting. In later papers, continuous densities [6], rougher measures [8], more general cost functions (albeit still close to the quadratic cost functional) [21], as well as almost-minimisers of the quadratic cost functional [21] were considered. The quadratic version of Theorem 1.1 was also used to provide a more refined linearisation result of (1.2) in the quadratic set-up in [8] and of a similar statement in the context of optimal matching in [7].…”