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

In the following paper, one studies, given a bounded, connected open set $Ω$ $\subseteq$ R n , $κ$ > 0, a positive Radon measure $μ$ 0 in $Ω$ and a (signed) Radon measure $μ$ on $Ω$ satisfying $μ$($Ω$) = 0 and |$μ$| $κ$$μ$ 0 , the possibility of solving the equation div u = $μ$ by a vector field u satisfying |u| $κ$w on $Ω$ (where w is an integrable weight only related to the geometry of $Ω$ and to $μ$ 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 $Ω$, 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.