Solvability In Weighted Lebesgue Spaces of the Divergence Equation with Measure Data

Abstract

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, a positive Radon measure $\mu$ 0 in $\Omega$ and a (signed) Radon measure $\mu$ on $\Omega$ satisfying $\mu$($\Omega$) = 0 and |$\mu$| $\kappa$$\mu$ 0 , the possibility of solving the equation div u = $\mu$ by a vector field u satisfying |u| $\kappa$w on $\Omega$ (where w is an integrable weight only related to the geometry of $\Omega$ and to $\mu$ 0), together with a mild boundary condition. This extends results obtained in [4] for the equation div u = f , improving them on two aspects: one works here with the divergence equation with measure data, and also construct a weight w that relies in a softer way on the geometry of $\Omega$, improving its behavior (and hence the a priori behavior of the solution we construct) substantially in some instances. The method used in this paper follows a constructive approach of Bogovskii type.