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A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

A. Colesanti, M. Focardi, P. Guan, P. Salani

Abstract

We study the mixed Christoffel problem for $C^{2,+}$ convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a $C^{2,+}$ convex body.

A constant rank theorem for linear elliptic equations on the sphere with applications to the mixed Christoffel problem

Abstract

We study the mixed Christoffel problem for convex bodies providing sufficient conditions for its solution. Key to our approach is a constant rank theorem, following the approach developed in \cite{Guan-Ma-2003} to address the Christoffel problem, in order to ensure that the solution to a related second order linear PDE on the sphere is indeed geometric, that is, it is the support functions of a convex body.

Paper Structure

This paper contains 12 sections, 10 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. The passage to PDE's
  3. A sufficient condition for the Christoffel problem
  4. The mixed Christoffel problem
  5. The mixed Christoffel problem in the $3$-dimensional space
  6. The case $n=2$
  7. Proof of Theorem \ref{['main theorem']} in the case $n=2$
  8. Proof of Theorems \ref{['main theorem n=2 rephrased']}, \ref{['MC 3dim 1']} and \ref{['MC 3dim']}
  9. The $n$-dimensional case
  10. Some preliminary facts
  11. The differential inequality for the function $\phi$
  12. Proof of Corollaries \ref{['the corollary']} and \ref{['the corollary nge3']}

Key Result

Theorem A

Let $f\in C^{1,1}({{\mathbb S}^{n}})$ be a strictly positive function satisfying (eq condition in GM). Assume moreover that If $u$ is a solution to CF-eq with then $W(x)>0$ for every $x\in\mathbb S^n$.

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem A
  • Remark 1.3
  • Theorem 1.1
  • Definition 1.4
  • Theorem 1.2
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 1.3
  • Remark 1.7
  • ...and 21 more