Pure and Applied Mathematics Quarterly

Volume 17 (2021)

Number 3

Special Issue in Honor of Duong H. Phong

Edited by Tristan Collins, Valentino Tosatti, and Ben Weinkove

A class of curvature type equation

Pages: 865 – 907

DOI: https://dx.doi.org/10.4310/PAMQ.2021.v17.n3.a2

Authors

Pengfei Guan (Department of Mathematics and Statistics, McGill University, Montreal, Canada)

Xiangwen Zhang (Department of Mathematics, University of California, Irvine, Calif., U.S.A.)

Abstract

In this paper, we study the solvability of a general class of fully nonlinear curvature equations, which can be viewed as generalizations of the equations for Christoffel–Minkowski problem in convex geometry. We will also study the Dirichlet problem of the corresponding degenerate equations as an extension of the equations studied by Krylov.

Keywords

curvature equations, $C^{1,1}$ estimates

2010 Mathematics Subject Classification

Primary 53C21, 53C45. Secondary 35J60.

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Research of the first author was supported in part by an NSERC Discovery Grant. Research of the second author was supported by the NSF under Grant DMS-1809582.

Received 22 March 2019

Accepted 22 August 2019

Published 14 June 2021