$L^p$ Christoffel-Minkowski problem: the case $1< p<k+1$

TL;DR

This paper studies a nonlinear PDE related to the intermediate $L^p$ Christoffel-Minkowski problem for $1<p<k+1$, establishing existence of convex bodies with prescribed measures and regularity results under certain conditions.

Contribution

It proves the existence of convex bodies with prescribed $k$-th $p$-area measures for $1<p<k+1$ and provides regularity estimates for solutions when $p eq rac{k+1}{2}$.

Findings

Existence of convex bodies with prescribed measures under certain conditions.

Construction of examples showing geometric conditions are necessary.

Regularity estimates for solutions when $p eq rac{k+1}{2}$.

Abstract

We consider a fully nonlinear partial differential equation associated to the intermediate $L^p$ Christoffel-Minkowski problem in the case $1<p<k+1$. We establish the existence of convex body with prescribed $k$-th even $p$-area measure on $\mathbb S^n$, under an appropriate assumption on the prescribed function. We construct examples to indicate certain geometric condition on the prescribed function is needed for the existence of smooth strictly convex body. We also obtain $C^{1,1}$ regularity estimates for admissible solutions of the equation when $ p\ge \frac{k+1}2$.