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

Pierre-Étienne Druet

doi:10.3934/dcdss.2015.8.475

Article Contents

2015, Volume 8, Issue 3: 475-496. Doi: 10.3934/dcdss.2015.8.475

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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

Pierre-Étienne Druet^1,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin

Received: September 30, 2013

Revised: May 31, 2014

Published: June 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 35D10, 35J55, 35Q60.

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