On the structure of diffuse measures for parabolic capacities

Klimsiak, Tomasz; Rozkosz, Andrzej

doi:10.1016/j.crma.2019.04.012

Partial differential equations/Potential theory

On the structure of diffuse measures for parabolic capacities
[Sur la structure des mesures diffuses des capacités paraboliques]

Présenté par : Haïm Brézis

Klimsiak, Tomasz ¹ ; Rozkosz, Andrzej ¹

¹ Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Comptes Rendus. Mathématique, Tome 357 (2019) no. 5, pp. 443-449.

Résumé
Abstract

Soit $Q = (0, T) \times Ω$ , où Ω est un ouvert borné dans $R^{d}$ . 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 $μ = f + div (G) + g_{t}$ , où $f \in L^{1} (Q)$ , $G \in {(L^{p^{'}} (Q))}^{d}$ et $g \in L^{p} (0, T; W_{0}^{1, p} (Ω) \cap L^{2} (Ω))$ . Nous montrons l'inverse de ce résultat : si $p > 1$ , alors toute mesure Radon bornée qui admet une telle décomposition est diffuse.

Let $Q = (0, T) \times Ω$ , where Ω is a bounded open subset of $R^{d}$ . 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 $μ = f + div (G) + g_{t}$ for some $f \in L^{1} (Q)$ , $G \in {(L^{p^{'}} (Q))}^{d}$ and $g \in L^{p} (0, T; W_{0}^{1, p} (Ω) \cap L^{2} (Ω))$ . We show the converse of this result: if $p > 1$ , then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.

Reçu le : 2019-04-29
Accepté le : 2019-04-30
Publié le : 2019-05-01

DOI : 10.1016/j.crma.2019.04.012

Affiliations des auteurs :

Klimsiak, Tomasz ¹ ; Rozkosz, Andrzej ¹

¹ Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

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     title = {On the structure of diffuse measures for parabolic capacities},
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Klimsiak, Tomasz; Rozkosz, Andrzej. On the structure of diffuse measures for parabolic capacities. Comptes Rendus. Mathématique, Tome 357 (2019) no. 5, pp. 443-449. doi : 10.1016/j.crma.2019.04.012. http://www.numdam.org/articles/10.1016/j.crma.2019.04.012/

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Cité par Sources :

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