The $L_p$ dual Christoffel-Minkowski problem for $1<p<q\le k+1$ with $1\le k\le n$

Abstract

In this paper, we investigate an $L_p$ Christoffel-Minkowski-type problem that prescribes a class of $L_p$ geometric measures, which are mixtures of the $k$-th area measure and the $q$-th dual curvature measure. By establishing a gradient estimate, we obtain the existence of an even, smooth, strictly convex solution to this problem for $1<p<q\le k+1$, where $1\le k\le n$ and $n\ge 1$.