Let , where Ω is a bounded open subset of . 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, , then it is of the form for some , and . We show the converse of this result: if , then each bounded Radon measure μ on Q admitting such a decomposition is diffuse.
Soit , où Ω est un ouvert borné dans . 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 , où , et . Nous montrons l'inverse de ce résultat : si , alors toute mesure Radon bornée qui admet une telle décomposition est diffuse.
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Tomasz Klimsiak 1; Andrzej Rozkosz 1
@article{CRMATH_2019__357_5_443_0, author = {Tomasz Klimsiak and Andrzej Rozkosz}, title = {On the structure of diffuse measures for parabolic capacities}, journal = {Comptes Rendus. Math\'ematique}, pages = {443--449}, publisher = {Elsevier}, volume = {357}, number = {5}, year = {2019}, doi = {10.1016/j.crma.2019.04.012}, language = {en}, }
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/
[1] Nonlinear elliptic equations with right-hand side measures, Commun. Partial Differ. Equ., Volume 17 (1992), pp. 641-655
[2] Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 13 (1996), pp. 539-551
[3] Nonlinear elliptic equations with measures revisited (J. Bourgain; C. Kenig; S. Klainerman, eds.), Mathematical Aspects of Nonlinear Dispersive Equations, Annals of Mathematics Studies, vol. 163, Princeton University Press, Princeton, NJ, USA, 2007, pp. 55-110
[4] Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4), Volume 28 (1999), pp. 741-808
[5] Parabolic capacity and soft measures for nonlinear equations, Potential Anal., Volume 19 (2003), pp. 99-161
[6] On the closable parts of pre-Dirichlet forms and the fine supports of underlying measures, Osaka J. Math., Volume 28 (1991), pp. 517-535
[7] On the structure of bounded smooth measures associated with a quasi-regular Dirichlet form, Bull. Pol. Acad. Sci., Math., Volume 65 (2017), pp. 45-56
[8] Stability properties, existence, and nonexistence of renormalized solutions for elliptic equations with measure data, Commun. Partial Differ. Equ., Volume 27 (2002), pp. 2267-2310
[9] Renormalized solutions of nonlinear parabolic equations with general measure data, Ann. Mat. Pura Appl., Volume 187 (2008), pp. 563-604
[10] Approximation of diffuse measures for parabolic capacities, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008), pp. 161-166
[11] Diffuse measures and nonlinear parabolic equations, J. Evol. Equ., Volume 11 (2011), pp. 861-905
[12] On the notion of renormalized solution to nonlinear parabolic equations with general measure data, J. Elliptic Parabolic Equ., Volume 1 (2015), pp. 201-214
[13] Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., Volume 14 (1983), pp. 522-533
[14] Compact sets in the space , Ann. Mat. Pura Appl., Volume 146 (1987), pp. 65-96
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☆ Research supported by Polish National Science Centre (Grant No. 2016/23/B/ST1/01543).
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