Dual Curvature Density Equation with Group Symmetry

Abstract: This paper studies the general Lp dual curvature density equation under a group symmetry assumption. This geometric partial differential equation arises from the general Lp dual Minkowski problem of prescribing the Lp dual curvature measure of convex bodies. It is a Monge-Ampere 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 Lp dual Minkowski problem of origin-symmetric convex bodies.