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Article Dans Une Revue Studia Mathematica Année : 2021

SOLVABILITY IN WEIGHTED LEBESGUE SPACES OF THE DIVERGENCE EQUATION WITH MEASURE DATA

Laurent Moonens
Emmanuel Russ

Résumé

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.
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Dates et versions

hal-02508132 , version 1 (13-03-2020)

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Laurent Moonens, Emmanuel Russ. SOLVABILITY IN WEIGHTED LEBESGUE SPACES OF THE DIVERGENCE EQUATION WITH MEASURE DATA. Studia Mathematica, 2021, 259 (3), pp.305-326. ⟨hal-02508132⟩
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