Quantitative Linearization Results for the Monge‐Ampère Equation

Abstract

This paper is about quantitative linearization results for the Monge-Ampère equation with rough data. We develop a large-scale regularity theory and prove that if a measure µ is close to the Lebesgue measure in Wasserstein distance at all scales, then the displacement of the macroscopic optimal coupling is quantitatively close at all scales to the gradient of the solution of the corresponding Poisson equation. The main ingredient we use is a harmonic approximation result for the optimal transport plan between arbitrary measures. This is used in a Campanato iteration which transfers the information through the scales.

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

“…This provides a keystone to the line of research started in [13] and continued in [14]: In [13], the variational approach was introduced and -regularity was established in case of a Euclidean cost function c(x, y) = 1 2 |x − y| 2 , see [13,Theorem 1.2]. In [14], among other things, the argument was extended to rougher densities, which required a substitute for McCann's displacement convexity; this generalization is crucial here.…”

“…In this paper, we give an entirely variational proof of the -regularity result for optimal transportation with a general cost function c and Hölder continuous densities, as established by De Philippis and Figalli [7]. This provides a keystone to the line of research started in [13] and continued in [14]: In [13], the variational approach was introduced and -regularity was established in case of a Euclidean cost function c(x, y) = 1 2 |x − y| 2 , see [13,Theorem 1.2]. In [14], among other things, the argument was extended to rougher densities, which required a substitute for McCann's displacement convexity; this generalization is crucial here.…”

“…• Both approaches have to establish an approximate orthogonality that allows to relate the distance between the minimal configuration and the construction in an energy norm by the energy gap, see [21, p.426ff.] and [13, Proof of Proposition 3.3, Step 3] in the simple setting or rather [14,Lemma 1.8] in our setting; it thus ultimately relies on some (strict) convexity, see [21, (4)]. • In order to establish this approximate orthogonality, both approaches have to smooth out the boundary data (there by simple convolution, here in addition by a nonlinear approximation), see [21, (34), (40), (52)] and [14,Proposition 3.6,(3.46), (3.47)].…”

“…and [13, Proof of Proposition 3.3, Step 3] in the simple setting or rather [14,Lemma 1.8] in our setting; it thus ultimately relies on some (strict) convexity, see [21, (4)]. • In order to establish this approximate orthogonality, both approaches have to smooth out the boundary data (there by simple convolution, here in addition by a nonlinear approximation), see [21, (34), (40), (52)] and [14,Proposition 3.6,(3.46), (3.47)]. • In view of this, both approaches have to choose a good radius for the cylinder (in the Eulerian space-time here) on which the construction is carried out, see [21, p.424] and [14,Section 3.1.3].…”

“…7 In our case, this simple concept means that, on a given scale, we interpret the minimizer (always with respect to its own boundary conditions) of the problem with c as an approximate minimizer of the Euclidean problem. This allows us to directly appeal to the Euclidean harmonic approximation [14,Theorem 1.5]. Incidentally, while dealing with Hölder continuous densities like in [13] and not general measures as in [14], we could not appeal to the simpler [13,Proposition 3.3], since this one relies on the Euler-Lagrange equation in form of McCann's displacement convexity.…”