Geodesic
Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods
Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 645-661.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

We consider the homogeneous Dirichlet problem for nonlinear elliptic equations as \begin{equation*}-\operatorname{div} a(x, \nabla u) = b(x, \nabla u) + \mu \end{equation*} where $\mu$ is a measure with bounded total variation. We fix structural conditions on functions $a$, $b$ which ensure existence of solutions. Moreover, if $\mu$ is an $L^1$ function, we prove a uniqueness result under more restrictive hypotheses on the operator. 
@article{BUMI_2008_9_1_3_a7,
     author = {Alvino, Angelo and Mercaldo, Anna},
     title = {Nonlinear {Elliptic} {Equations} with {Lower} {Order} {Terms} and {Symmetrization} {Methods}},
     journal = {Bollettino della Unione matematica italiana},
     pages = {645--661},
     publisher = {mathdoc},
     volume = {Ser. 9, 1},
     number = {3},
     year = {2008},
     zbl = {1191.35125},
     mrnumber = {2455337},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a7/}
}
TY  - JOUR
AU  - Alvino, Angelo
AU  - Mercaldo, Anna
TI  - Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods
JO  - Bollettino della Unione matematica italiana
PY  - 2008
SP  - 645
EP  - 661
VL  - 1
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a7/
LA  - en
ID  - BUMI_2008_9_1_3_a7
ER  - 
%0 Journal Article
%A Alvino, Angelo
%A Mercaldo, Anna
%T Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods
%J Bollettino della Unione matematica italiana
%D 2008
%P 645-661
%V 1
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a7/
%G en
%F BUMI_2008_9_1_3_a7
Alvino, Angelo; Mercaldo, Anna. Nonlinear Elliptic Equations with Lower Order Terms and Symmetrization Methods. Bollettino della Unione matematica italiana, Série 9, Tome 1 (2008) no. 3, pp. 645-661. http://geodesic.mathdoc.fr/item/BUMI_2008_9_1_3_a7/

[1] A. Alvino - A. Mercaldo, Nonlinear elliptic problems with $L^1$ data: an approach via symmetrization methods, Mediterr. J. Math. (to appear) | DOI | MR | Zbl

[2] A. Alvino - G. Trombetti, Su una classe di equazioni ellittiche degeneri, Ricerche Mat., 29 (1980), 193-212. | MR

[3] Ph. Bénilan - L. Boccardo - Th. Gallouët - R. Gariepy - M. Pierre, - J.L. Vázquez, An $L^1$ theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273. | fulltext EuDML | MR

[4] M.F. Betta - A. Mercaldo - F. Murat - M.M. Porzio, Uniqueness of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side in $L^1$. A tribute to J.L. Lions. (electronic), ESAIM Control Optim. Calc. Var., 8 (2002), 239-272. | fulltext EuDML | DOI | MR | Zbl

[5] M.F. Betta - A. Mercaldo - F. Murat - M.M. Porzio, Existence of renormalized solutions to nonlinear elliptic equations with a lower order term and right-hand side measure, J. Math. Pures Appl., 82 (2003), 90-124. | DOI | MR | Zbl

[6] M.F. Betta - V. Ferone - A. Mercaldo, Regularity for solutions of nonlinear elliptic equations, Bull. Sci. Math., 118 (1994), 539-567. | MR | Zbl

[7] L. Boccardo - Th. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. | DOI | MR

[8] L. Boccardo - T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations, 17 (1992), 641-655. | DOI | MR | Zbl

[9] J. Casado-Diaz - F. Murat - A. Porretta, Uniqueness results for pseudomonotone problems with $p > 2$, C.R. Acad. Sci. Paris, Ser. I, 344(2007), 487-492. | DOI | MR | Zbl

[10] A. Dall'Aglio, Approximated solutions of equations with $L^1$ data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), 207-240. | DOI | MR

[11] G. Dal Maso - F. Murat - L. Orsina - A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Sc. Norm. Sup. Pisa Cl. Sci. , (4) 28, (1999), 741-808. | fulltext EuDML | MR | Zbl

[12] T. Del Vecchio, Nonlinear elliptic equations with measure data, Potential Anal., 4 (1995), 185-203. | DOI | MR | Zbl

[13] T. Del Vecchio - M.M. Porzio, Existence results for a class of non coercive Dirichlet problems, Ricerche Mat., 44 (1995), 421-438. | MR | Zbl

[14] A. Ferone - R. Volpicelli, Some relations between pseudo-rearrangement and relative rearrangement, Nonlinear Anal., 41, (2000), 855-869. | DOI | MR | Zbl

[15] G.H. Hardy - J.E. Littlewood - G. Polya, Inequalities, Cambridge University Press, Cambridge, (1964). | MR

[16] R. Hunt, On $L(p, q)$ spaces, Enseignement Math., 12 (1966), 249-276. | MR

[17] M. A. Krasnosel'Skii, Topologicol methods in the theory of nonlinear integral equations, Pergamon Press, New York, (1964). | MR

[18] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod et Gauthier-Villars, Paris, (1969). | MR | Zbl

[19] P.L. Lions - F. Murat, Sur les solutions renormalisées d'équations elliptiques non linéaires, manuscript.

[20] V.G. Maz'Ya, On weak solutions of the Dirichlet and Neumann problems, Trans. Moscow Math. Soc., 20 (1969), 135-172.

[21] F. Murat, Équations elliptiques non linéaires avec second membre $L^1$ ou mesure, Actes du 26ème Congrés National d'Analyse Numérique, Les Karellis, France (1994), A12-A24.

[22] R. O'Neil, Integral transform and tensor products on Orlicz spaces and $L(p,q)$ spaces, J. Analyse Math., 21 (1968), 1-276. | MR

[23] J.M. Rakotoson - R. Temam, Relative rearrangement in quasilinear elliptic variational inequalities, Indiana Univ. Math. J., 36 (1987), 757-810. | DOI | MR

[24] J. Serrin, Pathological solution of elliptic differential equations, Ann. Scuola. Norm. Sup. Pisa, (3) 18 (1964), 385-387. | fulltext EuDML | MR | Zbl

[25] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier (Grenoble), 15 (1965), 189-258. | fulltext EuDML | MR | Zbl

[26] G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl., 120 (1979), 160-184. | DOI | MR | Zbl

[27] G. Talenti, Linear elliptic P.D.E.'s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat., 4-B (1985), 917-949. | MR | Zbl