Propagation of minima for nonlocal operators

TL;DR

This paper characterizes the geometry of minima sets for supersolutions of fully nonlinear nonlocal equations involving fractional truncated Laplacians and eigenvalues, revealing sharp maximum principles.

Contribution

It introduces a sharp maximum principle for supersolutions of nonlocal operators with a $k$-dimensional nonlocality, advancing understanding of their minima sets.

Findings

Characterization of minima sets for supersolutions

Sharp maximum principle established for nonlocal operators

Insights into the geometry of solutions for fractional operators

Abstract

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-\emph{th fractional truncated Laplacian} or the $k$-\emph{th fractional eigenvalue} which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.