Comptes Rendus
Partial Differential Equations
A new concept of reduced measure for nonlinear elliptic equations
Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 169-174.

We study the existence of solutions of the nonlinear problem

Δu+g(u)=μinΩ,u=0onΩ,(i)
where μ is a Radon measure and g:RR is a nondecreasing continuous function with g(t)=0, t0. 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=0onΩ,(ii)
μ est une mesure de Radon et g est une fonction croissante et continue avec g(t)=0, t0. 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.

Received:
Published online:
DOI: 10.1016/j.crma.2004.05.012
Haïm Brezis 1, 2; Moshe Marcus 3; Augusto C. Ponce 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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Haïm Brezis; Moshe Marcus; Augusto C. Ponce. A new concept of reduced measure for nonlinear elliptic equations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 3, pp. 169-174. doi : 10.1016/j.crma.2004.05.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.05.012/

[1] P. Baras; M. Pierre Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble), Volume 34 (1984), pp. 185-206

[2] Ph. Bénilan; H. Brezis Nonlinear problems related to the Thomas–Fermi equation, J. Evol. Equations, Volume 3 (2004), pp. 673-770

[3] L. Boccardo; T. Gallouët; L. Orsina Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 13 (1996), pp. 539-551

[4] H. Brezis Some variational problems of the Thomas–Fermi type, Proc. Internat. School, Erice, 1978 (R.W. Cottle; F. Giannessi; J.-L. Lions, eds.), Wiley, Chichester (1980), pp. 53-73

[5] H. Brezis Nonlinear elliptic equations involving measures (C. Bardos; A. Damlamian; J.I. Diaz; J. Hernandez, eds.), Contributions to Nonlinear Partial Differential Equations, Madrid, 1981, Pitman, Boston, MA, 1983, pp. 82-89

[6] H. Brezis; A.C. Ponce Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 599-604

[7] H. Brezis, M. Marcus, A.C. Ponce, Nonlinear elliptic equations with measures revisited, in preparation

[8] H. Brezis; W.A. Strauss Semilinear second-order elliptic equations in L1, J. Math. Soc. Japan, Volume 25 (1973), pp. 565-590

[9] L. Dupaigne, A.C. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.), in press

[10] J.L. Vázquez On a semilinear equation in R2 involving bounded measures, Proc. Roy. Soc. Edinburgh Sect. A, Volume 95 (1983), pp. 181-202

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