Dual curvature density equation with group symmetry

Abstract

This paper studies the general $L_p$ dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general $L_p$ dual Minkowski problem of prescribing the $L_p$ dual curvature measure of convex bodies. It is a Monge-Ampère type equation on the unit sphere. If the density function of the dual curvature measure is invariant under a closed subgroup of the orthogonal group, the geometric partial differential equation is solved in this paper for certain range of negative p using a variational method. This work generalizes recent results on the $L_p$ dual Minkowski problem of origin-symmetric convex bodies.