Uniform ellipticity for nonlocal k-Hessian equations with k≥3

Establish uniform ellipticity of nonlocal k-Hessian equations for every integer k≥3 without assuming that the solutions are convex, using an admissibility condition natural for the corresponding k-Hessian equations.

Background

The paper establishes uniform ellipticity for the nonlocal 2-Hessian equation without imposing convexity of the solution, relying instead on a nonlocal semiconcavity hypothesis and strict positivity of the equation’s right-hand side. Earlier work had established strict ellipticity for nonlocal k-Hessian equations under a convexity assumption.

The authors identify extending this result to k≥3, while removing the convexity assumption, as an open problem. They state that the geometric techniques developed in the paper appear potentially applicable to this extension, but do not resolve it.