Mathematics – Analysis of PDEs
Scientific paper
2012-03-15
Mathematics
Analysis of PDEs
9 pages
Scientific paper
Let $\mu = e^{-V} \ dx$ be a probability measure and $T = \nabla \Phi$ be the optimal transportation mapping pushing forward $\mu$ onto a log-concave compactly supported measure $\nu = e^{-W} \ dx$. We present a simple approach to the regularity problem for the corresponding Monge-Amp{\`e}re equation $e^{-V} = \det D^2 \Phi \cdot e^{-W(\nabla \Phi)}$ in Besov spaces $W^{\gamma,1}_{loc}$. We prove that $D^2 \Phi \in W^{\gamma,1}_{loc}$ provided $e^{-V}$ belongs to an appropriate Besov class and $W$ is convex. In particular, $D^2 \Phi \in L^p_{loc}$ for some $p>1$. Our proof does not rely on the previously known regularity results.
Kolesnikov Alexander V.
Tikhonov Sergey Yu.
No associations
LandOfFree
Regularity of the Monge-Ampère equation in Besov's space does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Regularity of the Monge-Ampère equation in Besov's space, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Regularity of the Monge-Ampère equation in Besov's space will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-32032