Pogorelov estimates for semi-convex solutions of 𝑘-curvature equations

Chen, Xiaojuan; Tu, Qiang; Xiang, Ni

doi:10.1090/proc/16820

Pogorelov estimates for semi-convex solutions of $k$-curvature equations
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by Xiaojuan Chen, Qiang Tu and Ni Xiang;

Proc. Amer. Math. Soc. 152 (2024), 2923-2936

DOI: https://doi.org/10.1090/proc/16820

Published electronically: May 9, 2024

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Abstract:

In this paper, we consider $k$-curvature equations $\sigma _k(\kappa [M_u])=f(x,u,\nabla u)$ subject to $(k+1)$-convex Dirichlet boundary data instead of affine Dirichlet data of Sheng, Urbas, and Wang [Duke Math. J. 123 (2004), pp. 235–264]. By using the crucial concavity inequality for Hessian operator of Lu [Calc. Var. Partial Differential Equations 62 (2023), p.23], we derive Pogorelov estimates of semi-convex admissible solutions for these $k$-curvature equations.

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