Propagation of minima for nonlocal operators

<jats:p>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 <jats:inline-formula> <jats:alternatives> <jats:tex-math>$k$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210523000495_inline1.png" /> </jats:alternatives> </jats:inline-formula><jats:italic>-th fractional truncated Laplacian</jats:italic> or the <jats:inline-formula> <jats:alternatives> <jats:tex-math>$k$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210523000495_inline2.png" /> </jats:alternatives> </jats:inline-formula><jats:italic>-th fractional eigenvalue</jats:italic> which are fully nonlinear integral operators whose nonlocality is somehow <jats:inline-formula> <jats:alternatives> <jats:tex-math>$k$</jats:tex-math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0308210523000495_inline3.png" /> </jats:alternatives> </jats:inline-formula>-dimensional.</jats:p>