Fractional analogue of k-Hessian operators

TL;DR

This paper introduces a fractional analogue of k-Hessian operators, extending regularity results for the case k=2 by establishing strict ellipticity and applying a nonlocal Evans-Krylov theorem.

Contribution

It develops a fractional version of k-Hessian operators as concave envelopes of fractional linear operators, reproducing known regularity results for k=2.

Findings

Fractional 2-Hessian operator is strictly elliptic.

Solutions are proven to be classical under certain conditions.

Extends regularity theory to fractional k-Hessian operators.

Abstract

Applying ideas of fractional analogue of Monge-Amp\'ere operator by L. Caffarelli and F. Charro, we consider an analogue of fractional k-Hessian operators expressed as concave envelopes of fractional linear operators, and reproduce the same regularity results when k=2. Under the set up of global solutions prescribing data at infinity and global barriers, the key estimate is to prove that fractional 2-Hessian operator is strictly elliptic. Then we can apply nonlocal Evans-Krylov theorem to prove such solutions are classical.