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Pré-Publication, Document De Travail Année : 2023

Silent sources in $L^p$ and Helmholtz-type decompositions

Résumé

We study the Poisson-Laplace equation of the form ∆PM = divM in Rn with M ∈ [Lp(Rn)]n, 1 < p < ∞. This equation appear in inverse potential problems of electroencephalography (EEG), magnetoencephalography (MEG) or problems geomagnetism under the quasi-static assumption of Maxwell’s equations. We aim to characterise those non-zero vector-fields M with bounded support, say Ω, that generate a constant PM in each connected component of Rn\Ω, such M are said to be Ω-silent. When Ω is open and its boundary has measure zero, we managed to show M is Ω-silent if and only if M = ∇ψ + D with D divergence-free is such the extension of ∇ψ (D) by zero to Rn is a gradient (divergence-free) on Rn, that is, all Ω-silent M have Helmholtz decompositions. For a given M ∈ [Lp(Ω)]n, with Ω say a Sobolev extension domain, we managed to show that M = ∇ψ + D + |∇h|q−2∇h, 1/p + 1/q = 1, for h a harmonic function such that ∇h is orthogonal to both Ω-silent ∇ψ and D. This decomposition is a non-linear generalisation of the Helmholtz decomposition.
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Dates et versions

hal-03915548 , version 1 (29-12-2022)
hal-03915548 , version 2 (21-12-2023)

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  • HAL Id : hal-03915548 , version 2

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Laurent Baratchart, Juliette Leblond, Masimba Nemaire. Silent sources in $L^p$ and Helmholtz-type decompositions. 2023. ⟨hal-03915548v2⟩
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