An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations

Philippe Bénilan; Lucio Boccardo; Thierry Gallouët; Ron Gariepy; Michel Pierre; Juan Luis Vazquez

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (1995)

  • Volume: 22, Issue: 2, page 241-273
  • ISSN: 0391-173X

How to cite


Bénilan, Philippe, et al. "An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 22.2 (1995): 241-273. <>.

author = {Bénilan, Philippe, Boccardo, Lucio, Gallouët, Thierry, Gariepy, Ron, Pierre, Michel, Vazquez, Juan Luis},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {-Laplacian; existence; uniqueness; entropy condition},
language = {eng},
number = {2},
pages = {241-273},
publisher = {Scuola normale superiore},
title = {An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations},
url = {},
volume = {22},
year = {1995},

AU - Bénilan, Philippe
AU - Boccardo, Lucio
AU - Gallouët, Thierry
AU - Gariepy, Ron
AU - Pierre, Michel
AU - Vazquez, Juan Luis
TI - An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 1995
PB - Scuola normale superiore
VL - 22
IS - 2
SP - 241
EP - 273
LA - eng
KW - -Laplacian; existence; uniqueness; entropy condition
UR -
ER -


  1. [B1] Ph. Bénilan, Equations d'évolution dans un espace de Banach quelconque et applications. Thesis, Univ. Orsay, 1972. 
  2. [B2] Ph. Bénilan, On the p-Laplacian in L1. Conference at the Analysis Seminar, Dept. of Math., Univ. Wisconsin at Madison, January 1987. 
  3. [BBC] Ph. Bénilan - H. Brezis - M.G. Crandall, A semilinear equation in L1. Ann. Scuola Norm. Sup. Pisa Cl. Sci.2 (1975), 523-555. Zbl0314.35077MR390473
  4. [BC] Ph. Bénilan - M.G. Crandall, Completely accretive operators. In: "Semigroup Theory and Evolution Equations" (Ph. Clément et al. eds.), M. Dekker, 1991, 41-76. Zbl0895.47036MR1164641
  5. [BCP] Ph. Bénilan - M.G. Crandall - A. Pazy, Evolution Equations governed by accretive operators. To appear. 
  6. [BeW] Ph. Bénilan - P. Wittbold, Absorptions non linéaires. J. Funct. Anal.114 (1993), 59-96. Zbl0786.47043MR1220983
  7. [BG1] L. Boccardo - Th. Gallouët, Non linear elliptic and parabolic equations involving measure data. Jour. Funct. Anal.87 (1989), 149-169. Zbl0707.35060MR1025884
  8. [BG2] L. Boccardo - Th. Gallouët, Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differential Equations17 (1992), 641-655. Zbl0812.35043MR1163440
  9. [BGV] L. Boccardo - Th. Gallouët - J.L. Vazquez, Nonlinear elliptic equations in R N without growth restrictions on the data. Jour. Diff. Eqns.105 (1993), 334-363. Zbl0810.35021MR1240399
  10. [BGDM] L. Boccardo - D. Giachetti - J.I. Diaz - F. Murat, Existence of a solution for a weaker form of a nonlinear elliptic equation. In: "Recent advances in nonlinear elliptic and parabolic problems" (Nancy, 1988), Pitman Res. Notes Math. Ser. 208, Longman Sc. Tech., Harlow, 1989, 229-246. Zbl0703.35063MR1035010
  11. [Br] H. Brezis, Opérateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert. North Holland, Amsterdam, 1973. Zbl0252.47055
  12. [BS] H. Brezis - W. Strauss, Semilinear elliptic equations in L 1. Jour. Math. Soc. Japan25 (1973), 565-590. Zbl0278.35041MR336050
  13. [Bw] F.E. Browder, Pseudomonotone operators and nonlinear elliptic boundary-value problems on unbounded domains. Proc. Nat. Acad. Sci.74 (1977), 2659-2661. Zbl0358.35034MR445124
  14. [C] M.G. Crandall, An introduction to evolution governed by accretive operators. In: "Dynamical Systems, An International Symposium ", vol. 1 (L. Cesari et al. eds.), Academic Press, New York, 1976, 131-165. Zbl0339.35049MR636953
  15. [D] A. Dall' Aglio, Alcuni problemi relativi alla H -convergenza di equazioni ellittiche e paraboliche quasilineari. Doctoral Thesis, Roma1, 1992 (in Italian). 
  16. [DBH1] E. Dibenedetto - M.A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation. Trans. Amer. Math. Soc.314 (1989), 187-224. Zbl0691.35047MR962278
  17. [DBH2] E. Dibenedetto - M.A. Herrero, Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when 1 &lt; p &lt; 2. Arch. Rat. Mech. Anal.111 (1990), 225-290. Zbl0726.35066MR1066761
  18. [DL] R.J. Di Perna - P.L. Lions, On the Cauchy problem for Boltzman equations: Global existence and weak stability. Ann of Math.130 (1989), 321-366. Zbl0698.45010MR1014927
  19. [GT] D. Gilbarg - N. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, Berlin, 1983. Zbl0562.35001MR737190
  20. [K] M.A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford, 1965. Zbl0111.30303MR159197
  21. [Kr] S.N. Kruzhkov, First-order quasilinear equations in several independent variables. Mat. USSR-Sbornik10 (1970), 217-243. Zbl0215.16203
  22. [La] P. Lax, Hyperbolic systems of conservation laws, II. Comm. Pure Applied Maths10 (1957), 537-566. Zbl0081.08803MR93653
  23. [Li] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969. Zbl0189.40603MR259693
  24. [LL] J. Leray - J.L. Lions, Quelques résultats de Vishik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France93 (1965), 97-107. Zbl0132.10502MR194733
  25. [LM] P.L. Lions - F. Murat, Sur les solutions renormalisées d' équations elliptiques non linéaires, to appear. 
  26. [M] F. Murat, Soluciones renormalizadas de EDP elipticas no lineales. Publ. Laboratoire d'Analyse Numérique, Univ. Paris 6, R 93023 (1993) [in Spanish]. 
  27. [R1] J.M. Rakotoson, Resolution of the critical cases for problems with L1-data. Asymptotic Analysis6 (1993), 229-246. Zbl0793.35042MR1201197
  28. [R2] J.M. Rakotoson, Generalized solutions in a new type of sets for problems with measures as data. Diff. Int. Equations6 (1993), 27-36. Zbl0780.35047MR1190163
  29. [R3] J.M. Rakotoson, Uniqueness of renormalized solutions in a T-set for the L1-data problem and the link between various formulations. Indiana Univ. Math. J.432 (1994), 285-293. Zbl0805.35035MR1291535
  30. [S] J. Serrin, Pathological solution of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci.18 (1964), 385-387. Zbl0142.37601MR170094
  31. [Ta] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Annali Mat. Pura Appl.120 (1979), 159-184. Zbl0419.35041MR551065

Citations in EuDML Documents

  1. Fahd Karami, Compétition Réaction-Diffusion et comportement asymptotique d’un problème d’obstacle doublement non linéaire
  2. Albo Carlos Cavalheiro, Existence of entropy solutions for degenerate quasilinear elliptic equations in L 1
  3. François Murat, Équations elliptiques non linéaires monotones avec un deuxième membre L 1 ou mesure
  4. Lucio Boccardo, Integral inequalities and summability of solutions of some differential problems
  5. F. Andreu, N. Igbida, J. M. Mazón, J. Toledo, L 1 existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions
  6. Philippe Bénilan, Jose Carrillo, Petra Wittbold, Renormalized entropy solutions of scalar conservation laws
  7. Abdelali Sabri, Ahmed Jamea, Hamad Talibi Alaoui, Existence of entropy solutions to nonlinear degenerate parabolic problems with variable exponent and L 1 -data
  8. Chiara Leone, Stability results for obstacle problems with measure data
  9. Lucio Boccardo, Haïm Brezis, Some remarks on a class of elliptic equations with degenerate coercivity
  10. Tommaso Leonori, Bounded Solutions for Some Dirichlet Problems with L 1 ( Ω ) Data

NotesEmbed ?


You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.


Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.