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 \( p\ge \frac{k+1}{2}\).
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Communicated by J. Jost.
Research of the first author is supported in part by an NSERC Discovery grant, the research of the second author is supported in part by NSFC (Grant No. 11501480) and the Natural Science Foundation of Fujian Province of China (Grant No. 2017J06003). Part of this work was done while CX was visiting the department of mathematics and statistics at McGill University. He would like to thank the department for its hospitality.
