Approximation of diffuse measures for parabolic capacities

Francesco Petitta; Augusto C. Ponce; Alessio Porretta

doi:10.1016/j.crma.2007.12.002

Partial Differential Equations

Approximation of diffuse measures for parabolic capacities

Presented by: Haïm Brezis

Francesco Petitta ¹; Augusto C. Ponce ²; Alessio Porretta ³

¹ Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy
² Laboratoire de mathématiques et physique théorique (CNRS UMR 6083), Université de Tours, 37200 Tours, France
³ Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 161-166.

Abstract
Résumé

Given a parabolic cylinder $Q = (0, T) \times Ω$ , with $Ω \subset R^{N}$ , we consider the class of finite measures which do not charge sets of zero p-parabolic capacity in Q. We prove that such measures can be strongly approximated by measures which can be written as $v_{t} - Δ_{p} v$ with $v \in L^{p} (0, T; W_{0}^{1, p} (Ω)) \cap L^{\infty} (Q)$ . Estimates on the capacity of level sets of solutions of parabolic equations play a crucial role in our proof.

Étant donné un cylindre parabolique $Q = (0, T) \times Ω$ , avec $Ω \subset R^{N}$ , on considère la classe des mesures bornées sur Q qui ne chargent pas les ensembles de p-capacité nulle. Nous démontrons que ces mesures peuvent être approchées au sens fort par des mesures de la forme $v_{t} - Δ_{p} v$ avec $v \in L^{p} (0, T; W_{0}^{1, p} (Ω)) \cap L^{\infty} (Q)$ . Des estimations sur la capacité des ensembles de niveau des solutions d'équations paraboliques jouent un rôle crucial dans notre preuve.

Accepted: 2007-11-25
Published online: 2008-01-14

DOI: 10.1016/j.crma.2007.12.002

Author's affiliations:

Francesco Petitta ¹; Augusto C. Ponce ²; Alessio Porretta ³

¹ Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy
² Laboratoire de mathématiques et physique théorique (CNRS UMR 6083), Université de Tours, 37200 Tours, France
³ Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma, Italy

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     author = {Francesco Petitta and Augusto C. Ponce and Alessio Porretta},
     title = {Approximation of diffuse measures for parabolic capacities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {161--166},
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     year = {2008},
     doi = {10.1016/j.crma.2007.12.002},
     language = {en},
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Francesco Petitta; Augusto C. Ponce; Alessio Porretta. Approximation of diffuse measures for parabolic capacities. Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 161-166. doi : 10.1016/j.crma.2007.12.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.12.002/

References
Cited by

[1] L. Boccardo; A. Dall'Aglio; T. Gallouët; L. Orsina Nonlinear parabolic equations with measure data, J. Funct. Anal., Volume 147 (1997), pp. 237-258

[2] H. Brezis; A.C. Ponce Reduced measures for obstacle problems, Adv. Differential Equations, Volume 10 (2005), pp. 1201-1234

[3] J. Droniou; A. Porretta; A. Prignet Parabolic capacity and soft measures for nonlinear equations, Potential Anal., Volume 19 (2003), pp. 99-161

[4] J.-L. Lions Quelques méthodes de résolution des problèmes aux limites non linéaires, Paris, Dunod, 1969

[5] F. Petitta, A.C. Ponce, A. Porretta, Strong approximation of diffuse measures and nonlinear parabolic equations, in preparation

[6] M. Pierre Parabolic capacity and Sobolev spaces, SIAM J. Math. Anal., Volume 14 (1983), pp. 522-533

[7] A. Porretta Existence results for nonlinear parabolic equations via strong convergence of truncations, Ann. Mat. Pura Appl. (IV), Volume 177 (1999), pp. 143-172

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