Locally uniform ellipticity of the fractional Hessian operators

2025/11/13 by Ziyu Gan, Gan, Ziyu, Heming Jiao +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2511.10034

openalex publication_date 2025/11/13 · openalex created_date 2025/11/15 · openalex updated_date 2026/07/28

Abstract

In [1], Caffarelli-Charro introduced a fractional Monge-Ampère operator. Later, Wu [17] generalized it to a fractional analogue of k-Hessian operators and proved the strict ellipticity for k=2. In this paper, we introduce a fractional analogue of general Hessian operators and prove the stability. We also show that the fractional analogue k-Hessian operators defined in [17] are strictly elliptic with respect to convex solutions for all 2 ≤ k ≤ n. Furthermore, we provide a new proof for the case k=2 without the convexity condition.

Citations

• The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds
• Aleksandrov-Bakelman-Pucci Type Estimates For Integro-Differential\n Equations
• The Evans-Krylov theorem for non local fully non linear equations
• The Evans-Krylov theorem for nonlocal fully nonlinear equations
• The Dirichlet problem for nonlinear second order elliptic equations, III: Functions of the eigenvalues of the Hessian

Related

• Fractional analogue of k-Hessian operators
• Radial fractional Laplace operators and Hessian inequalities
• Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality
• Local solvability of the k-Hessian equations
• Improved regularity for a Hessian-dependent functional