Convex capillary hypersurfaces of prescribed curvature problem

2025/04/19 by Mei, Xinqun, Wang, Guofang, Weng, Liangjun · 3 citations
#35B65 Secondary: 35J60 #53C42 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Primary: 53C21

paper · doi:10.48550/arxiv.2504.14392

Abstract

In this paper, we study the prescribed k-th Weingarten curvature problem for convex capillary hypersurfaces in ℝⁿ⁺¹₊. This problem naturally extends the prescribed k-th Weingarten curvature problem for closed convex hypersurfaces, previously investigated by Guan-Guan in [19], to the capillary setting. We reformulate the problem as the solvability of a Hessian quotient equation with a Robin boundary condition on a spherical cap. Under a natural sufficient condition, we establish the existence of a strictly convex capillary hypersurface with the prescribed k-th Weingarten curvature. This also extends our recent work on the capillary Minkowski problem in [40].

Cited by

• Prescribed L_p curvature problem for convex capillary hypersurface
• The capillary Christoffel-Minkowski problem
• The capillary Orlicz-Minkowski problem

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• A simple proof of curvature estimate for convex solution of k-Hessian equation