Comptes Rendus
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

Haïm Brezis1, 2 ; Moshe Marcus3 ; Augusto C. Ponce1, 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, Volume 339 (2004) no. 3, pp. 169-174.

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.

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.

Reçu le :
Publié le :
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 
@article{CRMATH_2004__339_3_169_0,
     author = {Ha{\"\i}m Brezis and Moshe Marcus and Augusto C. Ponce},
     title = {A new concept of reduced measure for nonlinear elliptic equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {169--174},
     publisher = {Elsevier},
     volume = {339},
     number = {3},
     year = {2004},
     doi = {10.1016/j.crma.2004.05.012},
     language = {en},
}
TY  - JOUR
AU  - Haïm Brezis
AU  - Moshe Marcus
AU  - Augusto C. Ponce
TI  - A new concept of reduced measure for nonlinear elliptic equations
JO  - Comptes Rendus. Mathématique
PY  - 2004
SP  - 169
EP  - 174
VL  - 339
IS  - 3
PB  - Elsevier
DO  - 10.1016/j.crma.2004.05.012
LA  - en
ID  - CRMATH_2004__339_3_169_0
ER  - 
%0 Journal Article
%A Haïm Brezis
%A Moshe Marcus
%A Augusto C. Ponce
%T A new concept of reduced measure for nonlinear elliptic equations
%J Comptes Rendus. Mathématique
%D 2004
%P 169-174
%V 339
%N 3
%I Elsevier
%R 10.1016/j.crma.2004.05.012
%G en
%F CRMATH_2004__339_3_169_0
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

  • Tomasz Klimsiak Nonlinear elliptic equations with self-adjoint integro-differential operators and measure data under sign condition on the nonlinearity, Advanced Nonlinear Studies, Volume 25 (2025) no. 1, p. 171 | DOI:10.1515/ans-2023-0153
  • Lucas C. F. Ferreira; Wender S. Lagoin On a localization-in-frequency approach for a class of elliptic problems with singular boundary data, Proceedings of the Royal Society of Edinburgh: Section A Mathematics (2024), p. 1 | DOI:10.1017/prm.2024.61
  • Debajyoti Choudhuri; Dušan D. Repovš; Kamel Saoudi A Double Phase Problem with a Nonlinear Boundary Condition, Bulletin of the Malaysian Mathematical Sciences Society, Volume 46 (2023) no. 4 | DOI:10.1007/s40840-023-01513-2
  • Tomasz Klimsiak Schrödinger equations with smooth measure potential and general measure data, Nonlinear Analysis, Volume 218 (2022), p. 112774 | DOI:10.1016/j.na.2021.112774
  • Chia-Yu Hsieh; Yong Yu Debye Layer in Poisson–Boltzmann Model with Isolated Singularities, Archive for Rational Mechanics and Analysis, Volume 236 (2020) no. 1, p. 289 | DOI:10.1007/s00205-019-01466-6
  • Shuibo Huang Quasilinear elliptic equations with exponential nonlinearity and measure data, Mathematical Methods in the Applied Sciences, Volume 43 (2020) no. 6, p. 2883 | DOI:10.1002/mma.6088
  • Tomasz Klimsiak Reduced measures for semilinear elliptic equations involving Dirichlet operators, Calculus of Variations and Partial Differential Equations, Volume 55 (2016) no. 4 | DOI:10.1007/s00526-016-1023-6
  • Christian Meyer; Lucia Panizzi; Anton Schiela Uniqueness Criteria for the Adjoint Equation in State-Constrained Elliptic Optimal Control, Numerical Functional Analysis and Optimization, Volume 32 (2011) no. 9, p. 983 | DOI:10.1080/01630563.2011.587074
  • Juan Luis Vázquez Measure-valued solutions and the phenomenon of blow-down in logarithmic diffusion, Journal of Mathematical Analysis and Applications, Volume 352 (2009) no. 1, p. 515 | DOI:10.1016/j.jmaa.2008.06.032
  • Louis Dupaigne; Augusto C. Ponce; Alessio Porretta Elliptic equations with vertical asymptotes in the nonlinear term, Journal d'Analyse Mathématique, Volume 98 (2006) no. 1, p. 349 | DOI:10.1007/bf02790280
  • Alberto Fiorenza Orlicz Capacities and Applications to PDEs and Sobolev Mappings, Elliptic and Parabolic Problems, Volume 63 (2005), p. 259 | DOI:10.1007/3-7643-7384-9_26
  • Augusto C. Ponce How to Construct Good Measures, Elliptic and Parabolic Problems, Volume 63 (2005), p. 375 | DOI:10.1007/3-7643-7384-9_37
  • J.L. Vázquez The Porous Medium Equation. New Contractivity Results, Elliptic and Parabolic Problems, Volume 63 (2005), p. 433 | DOI:10.1007/3-7643-7384-9_42
  • Haı¨m Brezis; Augusto C. Ponce Reduced measures on the boundary, Journal of Functional Analysis, Volume 229 (2005) no. 1, p. 95 | DOI:10.1016/j.jfa.2004.12.001

Cité par 14 documents. Sources : Crossref

Commentaires - Politique