On $T$-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data

A. A. Kovalevsky ; Yu. S. Gorban

Izvestiya. Mathematics , Tome 75 (2011) no. 1, pp. 101-156.

Résumé

We introduce the notions of $T$-solutions and shift $T$-solutions of variational inequalities corresponding to a non-linear degenerate anisotropic elliptic operator, a constraint set in a sufficiently large class, and an $L^1$-right-hand side. We prove theorems on the existence and uniqueness of such solutions and describe their properties. While the notion of $T$-solution is defined only when the constraint set contains at least one bounded function, the notion of shift $T$-solution does not require this condition. We describe the relation between these notions and prove that these types of solutions of a variational inequality coincide with ordinary solutions whenever the right-hand side is sufficiently regular.

Détail

Keywords: degenerate anisotropic elliptic variational inequalities, $L^1$-data, shift $T$-solution, existence and uniqueness of solutions.
Mots-clés : $T$-solution

Bibliographie

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

[2] D. Kinderlehrer, G. Stampacchia, An introduction to variational inequalities and their applications, Pure Appl. Math., 88, Academic Press, New York–London, 1980 | MR | MR | Zbl | Zbl

[3] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre, J. L. Vazquez, “An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22:2 (1995), 241–273 | MR | Zbl

[4] L. Boccardo, Th. Gallouët, “Nonlinear elliptic equations with right hand side measures”, Comm. Partial Differential Equations, 17:3–4 (1992), 641–655 | DOI | MR | Zbl

[5] L. Boccardo, Th. Gallouët, “Summability of the solutions of nonlinear elliptic equations with right hand side measures”, J. Convex Anal., 3:2 (1996), 361–365 | MR | Zbl

[6] L. Boccardo, Th. Gallouët, P. Marcellini, “Anisotropic equations in $L^1$”, Differential Integral Equations, 9:1 (1996), 209–212 | MR | Zbl

[7] L. Boccardo, Th. Gallouët, L. Orsina, “Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13:5 (1996), 539–551 | MR | Zbl

[8] F. Murat, “Soluciones renormalizadas de EDP elipticas no lineales”, Publ. Laboratoire d'Analyse Numérique, R93023, Univ. Paris VI, Paris, 1993

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

[10] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, “Definition and existence of renormalized solutions of elliptic equations with general measure data”, C. R. Acad. Sci. Paris Sér. I Math., 325:5 (1997), 481–486 | DOI | MR | Zbl

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

[12] 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 a measure”, J. Math. Pures Appl. (9), 82:1 (2003), 90–124 | DOI | MR | Zbl

[13] A. Alvino, V. Ferone, G. Trombetti, “Nonlinear elliptic equations with lower-order terms”, Differential Integral Equations, 14:10 (2001), 1169–1180 | MR | Zbl

[14] A. Alvino, L. Boccardo, V. Ferone, L. Orsina, G. Trombetti, “Existence results for nonlinear elliptic equations with degenerate coercivity”, Ann. Mat. Pura Appl. (4), 182:1 (2003), 53–79 | DOI | MR | Zbl

[15] A. Porretta, “Nonlinear equations with natural growth terms and measure data”, Proceedings of the 2002 Fez conference on partial differential equations, Electron. J. Differ. Equ. Conf., 9, Southwest Texas State Univ., San Marcos, TX, 2002, 183–202 | MR | Zbl

[16] S. Segura de León, “Existence and uniqueness for $L^1$ data of some elliptic equations with natural growth”, Adv. Differential Equations, 8:11 (2003), 1377–1408 | MR | Zbl

[17] A. A. Kovalevsky, “General conditions for limit summability of solutions of nonlinear elliptic equations with $L^1$-data”, Nonlinear Anal., 64:8 (2006), 1885–1895 | DOI | MR | Zbl

[18] A. A. Kovalevsky, “A priori properties of solutions of nonlinear equations with degenerate coercivity and $L^1$-data”, J. Math. Sci. (N. Y.), 149:5 (2008), 1517–1538 | DOI | MR

[19] L. Boccardo, Th. Gallouët, “Problèmes unilatéraux avec données dans $L^1$”, C. R. Acad. Sci. Paris Sér. I Math., 311:10 (1990), 617–619 | MR | Zbl

[20] L. Boccardo, G. R. Cirmi, “Existence and uniqueness of solution of unilateral problems with $L^1$ data”, J. Convex Anal., 6:1 (1999), 195–206 | MR | Zbl

[21] P. Oppezzi, A. M. Rossi, “Existence of solutions for unilateral problems with multivalued operators”, J. Convex Anal., 2:1–2 (1995), 241–261 | MR | Zbl

[22] P. Oppezzi, A. M. Rossi, “Esistenza di soluzioni per problemi unilateri con dato misura o in $L^1$”, Ricerche Mat., 45:2 (1996), 491–513 | MR | Zbl

[23] P. Oppezzi, A. M. Rossi, “Renormalized solutions for divergence problems with $L^1$ data”, Atti Sem. Mat. Fis. Univ. Modena, 46, suppl. (1998), 889–914 | MR | Zbl

[24] P. Oppezzi, M. Rossi, “Unilateral problems with measure data: links and convergence”, Differential Integral Equations, 14:9 (2001), 1051–1076 | MR | Zbl

[25] P. Oppezzi, A. M. Rossi, “Non coercive unilateral problems with lower order terms and measure data”, Ricerche Mat., 51:2 (2002), 357–379 | MR | Zbl

[26] Ch. Leone, “Existence and uniqueness of solutions for nonlinear obstacle problems with measure data”, Nonlinear Anal., Ser. A: Theory Methods, 43:2 (2001), 199–215 | DOI | MR | Zbl

[27] B. Brandolini, L. Randazzo, “An existence result for a class of variational inequalities with $L^1$ data”, Ricerche Mat., 50:2 (2001), 195–207 | MR | Zbl

[28] A. Benkirane, J. Bennouna, “Existence and uniqueness of solution of unilateral problems with $L^1$-data in Orlicz spaces”, Ital. J. Pure Appl. Math., 16 (2004), 87–102 | MR | Zbl

[29] L. Aharouch, Y. Akdim, E. Azroul, “Quasilinear degenerate elliptic unilateral problems”, Abstr. Appl. Anal., 2005, no. 1, 11–31 | DOI | MR | Zbl

[30] L. Aharouch, Y. Akdim, “Strongly nonlinear elliptic unilateral problems without sign condition and $L^1$ data”, J. Convex Anal., 13:1 (2006), 135–149 | MR | Zbl

[31] L. Aharouch, E. Azroul, M. Rhoudaf, “Nonlinear unilateral problems in Orlicz spaces”, Appl. Math. (Warsaw), 33:2 (2006), 217–241 | DOI | MR | Zbl

[32] A. Elmahi, D. Meskine, “Elliptic inequalities with lower order terms and $L^1$ data in Orlicz spaces”, J. Math. Anal. Appl., 328:2 (2007), 1417–1434 | DOI | MR | Zbl

[33] Y. Atik, J.-M. Rakotoson, “Local $T$-sets and degenerate variational problems. I”, Appl. Math. Lett., 7:4 (1994), 49–53 | DOI | MR | Zbl

[34] Y. Atik, “Local $T$-sets and degenerate quasilinear elliptic bilateral problems with an $L^1$-datum”, Nonlinear Anal., Ser. A: Theory Methods, 38:7 (1999), 827–867 | DOI | MR | Zbl

[35] L. Aharouch, E. Azroul, A. Benkirane, “Quasilinear degenerated equations with $L^1$ datum and without coercivity in perturbation terms”, Electron. J. Qual. Theory Differ. Equ., 2006, no. 19 | MR | Zbl

[36] A. A. Kovalevskii, “Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in $L^1$”, Izv. Math., 65:2 (2001), 231–283 | DOI | MR | Zbl

[37] A. A. Kovalevskii, “On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations”, Izv. Math., 67:5 (2003), 881–894 | DOI | MR | Zbl

[38] A. Kovalevsky, F. Nicolosi, “Entropy solutions of Dirichlet problem for a class of degenerate anisotropic fourth-order equations with $L^1$-right-hand sides”, Nonlinear Anal., Ser. A: Theory Methods, 50:5 (2002), 581–619 | DOI | MR | Zbl

[39] A. Kovalevsky, F. Nicolosi, “Solvability of Dirichlet problem for a class of degenerate anisotropic equations with $L^1$-right-hand sides”, Nonlinear Anal., 59:3 (2004), 347–370 | DOI | MR | Zbl

[40] A. Kovalevsky, F. Nicolosi, “Solvability of Dirichlet problem for a class of degenerate nonlinear high-order equations with $L^1$-data”, Nonlinear Anal., 47:1 (2001), 435–446 | DOI | MR | Zbl

[41] Yu. S. Gorban, “Suschestvovanie reshenii vyrozhdayuschikhsya anizotropnykh variatsionnykh neravenstv s $L^1$-pravymi chastyami”, Nelineinye granichnye zadachi, 13 (2003), 63–77 | Zbl

[42] A. A. Kovalevskii, Yu. S. Gorban, Vyrozhdayuschiesya anizotropnye variatsionnye neravenstva s $L^1$-dannymi, Preprint 2007.01, In-t prikladnoi matematiki i mekhaniki NAN Ukrainy, Donetsk, 2007

[43] A. A. Kovalevsky, Yu. S. Gorban, “Degenerate anisotropic variational inequalities with $L^1$-data”, C. R. Math. Acad. Sci. Paris, 345:8 (2007), 441–444 | DOI | MR | Zbl

[44] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Grundlehren Math. Wiss., 224, Springer-Verlag, Berlin, 1983 | MR | Zbl

[45] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974 | MR | MR | Zbl

[46] A. A. Kovalevskii, “O tochnom uslovii predelnoi summiruemosti reshenii nelineinykh ellipticheskikh uravnenii s $L^1$-pravymi chastyami”, Ukr. matem. visn., 2:4 (2005), 502–540 | MR | Zbl

[47] M. Troisi, “Teoremi di inclusione per spazi di Sobolev non isotropi”, Ricerche Mat., 18 (1969), 3–24 | MR | Zbl

[48] A. A. Kovalevskii, “Integrability of solutions of nonlinear elliptic equations with right-hand sides from classes close to $L^1$”, Math. Notes, 70:3 (2001), 337–346 | DOI | MR | Zbl

[49] A. A. Kovalevskii, “Integrability of solutions of nonlinear elliptic equations with right-hand sides from logarithmic classes”, Math. Notes, 74:5 (2003), 637–646 | DOI | MR | Zbl