Partial Differential Equations
Approximation of diffuse measures for parabolic capacities
Comptes Rendus. Mathématique, Volume 346 (2008) no. 3-4, pp. 161-166.

Given a parabolic cylinder Q=(0,T)×Ω, with ΩRN, 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 vtΔpv with vLp(0,T;W01,p(Ω))L(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)×Ω, avec ΩRN, 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 vtΔpv avec vLp(0,T;W01,p(Ω))L(Q). Des estimations sur la capacité des ensembles de niveau des solutions d'équations paraboliques jouent un rôle crucial dans notre preuve.

Accepted:
Published online:
DOI: 10.1016/j.crma.2007.12.002
Petitta, Francesco 1; Ponce, Augusto C. 2; Porretta, Alessio 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
@article{CRMATH_2008__346_3-4_161_0,
     author = {Petitta, Francesco and Ponce, Augusto C. and Porretta, Alessio},
     title = {Approximation of diffuse measures for parabolic capacities},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {161--166},
     publisher = {Elsevier},
     volume = {346},
     number = {3-4},
     year = {2008},
     doi = {10.1016/j.crma.2007.12.002},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.12.002/}
}
TY  - JOUR
AU  - Petitta, Francesco
AU  - Ponce, Augusto C.
AU  - Porretta, Alessio
TI  - Approximation of diffuse measures for parabolic capacities
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 161
EP  - 166
VL  - 346
IS  - 3-4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.12.002/
DO  - 10.1016/j.crma.2007.12.002
LA  - en
ID  - CRMATH_2008__346_3-4_161_0
ER  - 
%0 Journal Article
%A Petitta, Francesco
%A Ponce, Augusto C.
%A Porretta, Alessio
%T Approximation of diffuse measures for parabolic capacities
%J Comptes Rendus. Mathématique
%D 2008
%P 161-166
%V 346
%N 3-4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.12.002/
%R 10.1016/j.crma.2007.12.002
%G en
%F CRMATH_2008__346_3-4_161_0
Petitta, Francesco; Ponce, Augusto C.; Porretta, Alessio. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.12.002/

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

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

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

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

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

Cited by Sources: