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

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.

Higher integrability for the gradient of Mumford-Shah almost-minimizers

We extend a recent higher-integrability result for the gradient of minimizers of the Mumford-Shah functional to a suitable class of almost-minimizers. The extension crucially depends on an L∞ gradient estimate up to regular portions of the discontinuity set of an almost-minimizer.