no. 3
Partial Differential Equations
A new concept of reduced measure for nonlinear elliptic equations
[Un nouveau concept de mesure réduite pour des équations elliptiques non linéaires.]
Présenté par : Haïm Brezis
Brezis, Haïm1, 2 ; Marcus, Moshe3 ; Ponce, Augusto C.1, 2
1 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
2 Rutgers University, Department of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA
3 Technion, Department of Math., Haifa 32000, Israel
Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 169-174

We study the existence of solutions of the nonlinear problem

−Δu+g(u)=μinΩ,u=0on∂Ω,(i)
where μ is a Radon measure and g:R→R is a nondecreasing continuous function with g(t)=0, ∀t⩽0. Given g, Eq. (i) need not have a solution for every measure μ, and we say that μ is a good measure if (i) admits a solution. We show that for every μ there exists a largest good measure μ*⩽μ. This reduced measure μ* has a number of remarkable properties.

On étudie l'existence de solutions du problème non linéaire

−Δu+g(u)=μinΩ,u=0on∂Ω,(ii)
où μ est une mesure de Radon et g est une fonction croissante et continue avec g(t)=0, ∀t⩽0. Étant donné g, l'Éq. (ii) n'admet pas nécessairement de solution pour toute mesure μ. On dit que μ est une bonne mesure (relative à g) si (ii) admet une solution. On démontre que pour toute mesure μ, il existe une plus grande bonne mesure μ*⩽μ. La mesure réduite μ* a plusieurs propriétés remarquables.

Reçu le :
Publié le :
DOI : 10.1016/j.crma.2004.05.012

Brezis, Haïm 1, 2 ; Marcus, Moshe 3 ; Ponce, Augusto C. 1, 2

1 Laboratoire Jacques-Louis Lions, université Pierre et Marie Curie, BC 187, 4, pl. Jussieu, 75252 Paris cedex 05, France
2 Rutgers University, Department of Math., Hill Center, Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854, USA
3 Technion, Department of Math., Haifa 32000, Israel 
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Brezis, Haïm; Marcus, Moshe; Ponce, Augusto C. A new concept of reduced measure for nonlinear elliptic equations. Comptes Rendus. Mathématique, Tome 339 (2004) no. 3, pp. 169-174. doi: 10.1016/j.crma.2004.05.012

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