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Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations

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  • We show that $L^p$ vector fields over a Lipschitz domain are integrable to higher exponents if their generalized divergence and rotation can be identified with bounded linear operators acting on standard Sobolev spaces. A Div-Curl Lemma-type argument provides compact embedding results for such vector fields. We use these tools to investigate the regularity of the solution to the low-frequency approximation of the Maxwell equations in time-harmonic regime. We focus on the weak formulation `in H' of the problem, in a reference geometrical setting allowing for material heterogeneities and nonsmooth interfaces.
    Mathematics Subject Classification: Primary: 35D10, 35J55, 35Q60.


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