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.
Domaines
Mathématiques [math]Origine | Fichiers produits par l'(les) auteur(s) |
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