Regularity of linearized Monge-Ampere equation

Question

Consider the linearized Monge-Ampere equation $ (p-q)+\nabla\cdot(p\nabla\phi)=0$, where $p$ and $q$ are the density functions of two probability measures, which are supported on a bounded domain $\Omega$. The gradient of its solution $ \nabla \phi$ represents the optimal transport from probability $p$ to $q$ approximately. I am interested in the regularity of the solution $\phi$. In specific, under what regularity conditions on $p$ and $q$, the Sobolev norm $$\|\phi\|_{W^{k,p}(\Omega)}:=\big(\Sigma_{|\alpha|\le k}\|D^{\alpha}\phi\|^p_{L^p(\Omega)}\big)^{1/p}$$ can be upper bounded? Moreover, is it possible to show that the $\nabla^2 \phi$ is also Holder continuous and provide an explicit upper bound for the Holder continuity exponent?

Answer 1

Answers to questions like this (but for the not linearized version) can be found under "regularity of Kantorovich potentials". This is due to Brennier's theorem which says that the transport map $\phi$ is related to a Kantorovich potential (basically one of the two Lagrange multipliers of the optimal transport problem) by $u(x) = \phi(x) - \tfrac12\|x\|^2$ (holds for quadratic transport cost).

Some paper that have results on the regularity of these potentials are

Ma, Xi-Nan, Neil S. Trudinger, and Xu-Jia Wang. "Regularity of potential functions of the optimal transportation problem." Archive for rational mechanics and analysis 177.2 (2005): 151-183.

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Liu, Jiakun, Neil S. Trudinger, and Xu-Jia Wang. "Interior C 2, α regularity for potential functions in optimal transportation." Communications in Partial Differential Equations 35.1 (2009): 165-184.

or

Liu, Jiakun. "Hölder regularity of optimal mappings in optimal transportation." Calculus of Variations and Partial Differential Equations 34.4 (2009): 435.