On a Fractional Monge–Ampère Operator

Luis Caffarelli, Fernando Charro
2015 Annals of PDE  
In this paper we consider a fractional analogue of the Monge-Amp\'ere operator. Our operator is a concave envelope of fractional linear operators of the form $ \inf_{A\in \mathcal{A}}L_Au, $ where the set of operators corresponds to all affine transformations of determinant one of a given multiple of the fractional Laplacian. We set up a relatively simple framework of global solutions prescribing data at infinity and global barriers. In our key estimate, we show that the operator remains
more » ... ator remains strictly elliptic, which allows to apply known regularity results for uniformly elliptic operators and deduce that solutions are classical.
doi:10.1007/s40818-015-0005-x fatcat:2xg6atmbz5duthspnduon37xeq