| Abstract: |
| In this talk, I will present a fractional analogue of the classical $k$-Hessian operator, defined as an infimum of anisotropic fractional Laplacians over a suitable class of matrices.
I will focus on the case $k=2$ and show that, under natural assumptions, the fractional 2-Hessian operator is locally uniformly elliptic. This property places the operator within the framework of fully nonlinear integro-differential equations and makes available the regularity theory developed for nonlocal elliptic operators.
The key issue is to understand the possible degeneracies in the admissible class of matrices and to prove that these degenerate directions do not contribute to the infimum.
These results extend the theory known for the fractional Monge-Amp\`ere operator ($k=n$) developed by Caffarelli et al. in 2015.
This is ongoing joint work with Mar Gonz\`alez (UAM) and Fernando Charro (Wayne State). |
|