2020/03/16 by Moonens, Laurent, Russ, Emmanuel
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2003.07265
In the following paper, one studies, given a bounded, connected open set Ω ⊆ 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| κμ 0 , the possibility of solving the equation div u = μ by a vector field u satisfying |u| κw 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.