Locally uniform ellipticity of the fractional Hessian operators
Ziyu Gan, Heming Jiao
TL;DR
This work extends Hessian-type operators to a nonlocal fractional framework by introducing the fractional Hessian operator $F_s$ as a Bellman-type infimum of matrix-weighted fractional Laplacians. It establishes a precise nonlocal-to-local stability as $s\to1$, showing $\lim_{s\to1}(1-s)F_s[u](x)=\tfrac{\omega_n}{4}F(D^2u(x))$ for admissible, asymptotically linear functions, thereby linking the fractional and local Hessian operators. The authors prove strict ellipticity for fractional $k$-Hessian operators with $2\le k\le n$ under convexity and a positive lower bound on $(1-s)F_s[u]$, enabling nonlocal Evans–Krylov regularity $C^{2s+\alpha}$ for convex solutions and providing a convexity-free alternative for the case $k=2$. These results advance the theory of fully nonlinear nonlocal PDEs, offering a pathway to higher regularity for Hessian-type equations in a fractional setting and suggesting directions to remove convexity assumptions for larger $k$.
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 $2 \leq k \leq n$. Furthermore, we provide a new proof for the case $k=2$ without the convexity condition.