On divergence-free (form-bounded type) drifts

Abstract

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.