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

Abstract

In the present article, we prove the existence and uniqueness of smooth solutions to an anisotropic Lp{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 C2{C}^{2} bounds of solutions, the C1{C}^{1} bounds of solutions also need some efforts since the convexity of SS 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.

Keywords

• log-concave measure
• anisotropy
• lp minkowski problem
• monge-ampère equation
• the continuous method
• 35j96
• 53c42