A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic L p Minkowski problem for log-concave measure

Abstract

Abstract In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic L p {L}_{p} Minkowski problem for the log-concave measure. Our proof of the existence is based on the well-known continuous method whose crucial factor is the a priori bounds of an auxiliary problem. The uniqueness is based on a maximum principle argument. It is worth mentioning that apart from the C 2 {C}^{2} bounds of solutions, the C 1 {C}^{1} bounds of solutions also need some efforts since the convexity of S S cannot be used directly, which is one of great difference between the classical and the anisotropic versions. Moreover, our result can be seen as an attempt to get new results on the geometric analysis of log-concave measure.