The Harnack inequality for second-order elliptic equations with divergence-free drifts (Q471964)

The authors consider the elliptic equation \[ - \Delta u + b\cdot \nabla u +au =0 \] assuming \(b\) a divergence free vector field. Such kind of equation was studied by Stampacchia without the divergence free condition. Assuming the free boundary condition, it was studied by several authors, among them we quote Nazarov, Uralt'seva, Friedlander, Vicol, Seregin, Silvestre, Sverak and Zlatos. In this very interesting paper the authors establish a Harnack type inequality and a one side Liouville estimate under suitable assumptions of integrability on the coefficients \(a\) and \(b\).