General monotone formula for homogeneous k-Hessian equation in the exterior domain and its applications

2025/06/02 by Jiabin Yin, Yin, Jiabin, Xingjian Zhou +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.01434

openalex publication_date 2025/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we deal with an overdetermined problem for the k-Hessian equation (1≤ k<\frac n2) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \citeBNS. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to k-admissible solution u in \mathbb Rⁿ∖Ω, where Ω is smooth, k-convex and star-shaped domain, which constructed by Ma-Zhang\citeMZ and Xiao\citexiao.

Citations

• New monotonicity for p-capacitary functions in 3-manifolds with nonnegative scalar curvature
• The exterior Dirichlet problem for the homogeneous k-Hessian equation
• A geometric capacitary inequality for sub-static manifolds with harmonic\n potentials
• Minimising hulls, p-capacity and isoperimetric inequality on complete Riemannian manifolds
• Minkowski Inequalities via Nonlinear Potential Theory
• Symmetrization with respect to mixed volumes
• Sharp geometric inequalities for closed hypersurfaces in manifolds with nonnegative Ricci curvature
• Geometric aspects of p-capacitary potentials
• Radial symmetry for p-harmonic functions in exterior and punctured domains
• Classical Neumann Problems for Hessian equations and Alexandrov-Fenchel's inequalities
• Monotonicity formulas in potential theory
• An overdetermined problem for the anisotropic capacity
• On the geometry of the level sets of bounded static potentials
• Serrin-Type Overdetermined Problems: an Alternative Proof
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian
• A symmetry problem in potential theory
• Remark on the preceding paper of Serrin

Related

• The exterior Dirichlet problem for the homogeneous k-Hessian equation
• The Dirichlet problem of the homogeneous k-Hessian equation in a punctured domain
• Generalized Minkowski inequality via degenerate Hessian equations on exterior domains
• Hessian quotient equations on exterior domains
• A Serrin-type over-determined problem for Hessian equations in the exterior domain