Partial differential equations/Potential theory
On the structure of diffuse measures for parabolic capacities
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.

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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/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).