In this paper, we study the regularity of solutions to the Monge-Ampère equation. We prove the log-Lipschitz continuity for the gradient under certain assumptions. We also give a unified treatment for the continuity estimates of the second derivatives. As an application we show the local existence of continuous solutions to the semi-geostrophic equation arising in meteorology.
This article concerns the Monge-Ampère equations with infinite boundary value in convex domains in Euclidean space. We were able to characterize the growth rate conditions, which are nearly optimal, for the existence/nonexistence of solutions to the problem.
In this paper we show by example that there is no uniform estimate for the L p -Minkowski problem. As a result we obtain the nonuniqueness of solutions to the problem.
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