Degenerate elliptic equations with nonlinear boundary conditions and measures data

Andreu, Fuensanta; Igbida, Noureddine; Mazón, José M.; Toledo, Julián

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 767-803.

In this paper we study the questions of existence and uniqueness of solutions for equations of type $-\text{div}𝐚(x,Du)+\gamma \left(u\right)\ni {\mu }_1$, posed in an open bounded subset $\Omega$ of ${ℝ}^N$, with nonlinear boundary conditions of the form $𝐚(x,Du)·\eta +\beta \left(u\right)\ni {\mu }_2$. The nonlinear elliptic operator $\text{div}𝐚(x,Du)$ is modeled on the $p$-Laplacian operator ${\Delta }_p\left(u\right)=\text{div}\left({\left|Du\right|}^{p-2}Du\right)$, with $p>1$, $\gamma$ and $\beta$ are maximal monotone graphs in ${ℝ}^2$ such that $0\in \gamma \left(0\right)\cap \beta \left(0\right)$ and the data ${\mu }_1$ and ${\mu }_2$ are measures.

Classification : 35J60, 35D05

[1] F. Andreu, J. M. Mazón, S. Segura De León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Adv. Math. Sci. Appl. 7 (1997), 183–213. | MR | Zbl

[2] F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, $L^1$ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 61–89. | EuDML | Numdam | MR | Zbl

[3] F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces Free Bound. 8 (2006), 447–479. | MR | Zbl

[4] F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Obstacle problems for degenerate elliptic equations with nonlinear boundary conditions, Math. Models Methods Appl. Sci. 18 (2008), 1869–1893. | MR | Zbl

[5] P. Baras and M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), 185–206. | EuDML | Numdam | MR | Zbl

[6] Ph. Bénilan, L. Boccardo, Th. Gallouët, R. Gariepy, M. Pierre and 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. | EuDML | Numdam | MR | Zbl

[7] Ph. Bénilan and H. Brezis, Nonlinear problems related to the Thomas-Fermi equation, dedicated to Philippe Bénilan, J. Evol. Equ. 3 (2003), 673–770. | MR | Zbl

[8] Ph. Benilan, H. Brezis and M. G. Crandall, A semilinear equation in $L^1\left(R^N\right)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), 523–555. | EuDML | Numdam | MR | Zbl

[9] Ph. Bénilan, M. G. Crandall and P. Sacks, Some $L^1$ existence and dependence results for semilinear elliptic equations under nonlinear boundary conditions, Appl. Math. Optim. 17 (1986), 203–224. | MR | Zbl

[10] D. Bartolucci, F. Leoni, L. Orsina and A. C. Ponce, Semilinear equations with exponential nonlinearity and measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 799–815. | EuDML | Numdam | MR | Zbl

[11] L. Boccardo and T. Gallouet, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149–169. | MR | Zbl

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

[13] L. Boccardo, T. Gallouet and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 539–551. | EuDML | Numdam | MR | Zbl

[14] L. Boccardo and F. Murat, Remarques sur l’homogénéisation de certains problèmes quasi-linéaires. Portugaliae Math. 41 (1982), 535–562. | EuDML | MR | Zbl

[15] H. Brezis, Problèmes unilatéraux, J. Math. Pures Appl. 51 (1972), 1-168. | MR | Zbl

[16] H. Brezis, “Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert”, North-Holland, 1973. | MR | Zbl

[17] H. Brezis, “Analyse Fonctionnelle. Théorie et Applications”, Collection Mathématiques Appliquées pour la Maîtrise, Masson, Paris, 1983. | MR | Zbl

[18] H. Brezis, M. Marcus and A. C. Ponce, Nonlinear elliptic equations with measures revisited, In: “Mathematical Aspects of Nonlinear Dispersive Equations”, Ann. of Math. Stud., Vol. 163, Princeton Univ. Press, Princeton, NJ, 2007, 55–109. | MR | Zbl

[19] H. Brezis and A. C. Ponce, Reduced measures for obstacle problems, Adv. Differential Equations 10 (2005), 1201–1234. | MR | Zbl

[20] H. Brezis and W. Strauss, Semi-linear second-order elliptic equations in $L^1$, J. Math. Soc. Japan 25 (1973), 565-590. | MR | Zbl

[21] J. Crank, “Free and Moving Boundary Problems”, North-Holland, Amsterdam, 1977. | Zbl

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

[23] E. Dibenedetto and A. Friedman, The ill-posed Hele-Shaw model and the Stefan problem for supercooler water, Trans. Amer. Math. Soc. 282 (1984), 183–204. | MR | Zbl

[24] L. Dupaigne, A. C. Ponce and A. Porretta, Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math. 98 (2006), 349–396. | MR | Zbl

[25] G. Duvaux and J. L. Lions, “Inequalities in Mechanics and Physiscs”, Springer-Verlag, 1976. | MR

[26] C. M. Elliot and V. Janosky, A variational inequality approach to the Hele-Shaw flow with a moving boundary, Proc. Roy. Soc. Edinburg Sect. A 88 (1981), 93–107. | MR

[27] A. Gmira and L. Véron, Boundary singularities of solutions of some nonlinear elliptic equations. Duke Math. J. 64 (1991), 271–324. | MR | Zbl

[28] J. Heinonen, T. Kilpeläinen and O. Martio, “Nonlinear Potential Theory of Degenerate Elliptic Equations”, Oxford Mathematical Monographs, Oxford University Press, New York, 1993. | MR | Zbl

[29] J. L. Lions, “Quelques Méthodes de Résolution de Problémes aux Limites non Linéaires”, Dunod-Gauthier-Vilars, Paris, 1968. | MR

[30] M. Marcus and L. Véron, Removable singularities and boundary traces, J. Math. Pures Appl. 80 (2001), 879–900. | MR | Zbl

[31] M. Marcus and L. Véron, The precise boundary trace of solutions of a class of supercritical nonlinear equations, C. R. Math. Acad. Sci. Paris 344 (2007), 181–186. | MR | Zbl

[32] G. Stampacchia, Le probléme de Dirichlet pour les équations elliptiques du second order à coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258. | EuDML | Numdam | MR | Zbl

[33] J. L. Vázquez, On a semilinear equation in ${ℝ}^2$ involving bounded measures, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 181–202. | MR | Zbl

[34] W.P. Ziemer, “Weakly Differentiable Functions”, GTM 120, Springer-Verlag, 1989. | MR | Zbl