Comptes Rendus
no. 5
Partial differential equations/Potential theory
On the structure of diffuse measures for parabolic capacities
Presented by: Haïm Brézis
Tomasz Klimsiak 1 ; Andrzej Rozkosz 1
1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Comptes Rendus. Mathématique, Volume 357 (2019) no. 5, pp. 443-449

Let Q=(0,T)×Ω, where Ω is a bounded open subset of Rd. We consider the parabolic p-capacity on Q naturally associated with the usual p-Laplacian. Droniou, Porretta, and Prignet have shown that if a bounded Radon measure μ on Q is diffuse, i.e. charges no set of zero p-capacity, p>1, then it is of the form μ=f+div(G)+gt for some f∈L1(Q), G∈(Lp′(Q))d and g∈Lp(0,T;W01,p(Ω)∩L2(Ω)). We show the converse of this result: if p>1, then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.

Soit Q=(0,T)×Ω, où Ω est un ouvert borné dans Rd. On considère la p-capacité parabolique dans Q naturellement associée au p-laplacien. Droniou, Porretta et Prignet ont démontré que, si une mesure de Radon bornée μ dans Q est diffuse, c'est-à-dire si μ ne charge pas les ensembles de p-capacité nulle, elle est alors de la forme μ=f+div(G)+gt, où f∈L1(Q), G∈(Lp′(Q))d et g∈Lp(0,T;W01,p(Ω)∩L2(Ω)). Nous montrons l'inverse de ce résultat : si p>1, alors toute mesure Radon bornée qui admet une telle décomposition est diffuse.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.04.012

Tomasz Klimsiak  1 ; Andrzej Rozkosz  1

1 Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland 
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Tomasz Klimsiak; Andrzej Rozkosz. On the structure of diffuse measures for parabolic capacities. Comptes Rendus. Mathématique, Volume 357 (2019) no. 5, pp. 443-449. doi: 10.1016/j.crma.2019.04.012

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☆ Research supported by Polish National Science Centre (Grant No. 2016/23/B/ST1/01543).

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