Locally uniform ellipticity of the fractional Hessian operators

TL;DR

This paper introduces a fractional analogue of general Hessian operators, proves their stability and strict ellipticity for all relevant k-values, and offers a new proof for the k=2 case without convexity assumptions.

Contribution

It extends fractional Hessian operator theory by establishing stability and strict ellipticity for a broad class of these operators, including new proofs for specific cases.

Findings

Fractional Hessian operators are stable.

They are strictly elliptic for all 2 ≤ k ≤ n.

A new proof for the k=2 case without convexity.

Abstract

In [1], Caffarelli-Charro introduced a fractional Monge-Amp\`{e}re 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 $2 \leq k \leq n$. Furthermore, we provide a new proof for the case $k=2$ without the convexity condition.