On uniqueness of entropy solutions for nonlinear elliptic degenerate anisotropic
equations

Yu. S. Gorban

doi:10.15330/ms.47.1.59-70

In the present paper we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with $L^1$-right-hand sides in a bounded domain of ${\Bbb R}^n$ $(n \geqslant 2)$. This class is described by the presence of a set of exponents $q_1,\dots,q_n$ and a set of weighted functions $\nu_1,\dots,\nu_n$ in growth and coercitivity conditions on coefficients of the equations. The exponents $q_i$ characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions $\nu_i$ characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to study the uniqueness of entropy solution of the problem under consideration.

1. Aharouch L., Azroul E., Benkirane A. Quasilinear degenerated equations with $L^1$ datum and without coercivity in perturbation terms// Electron. J. Qual. Theory Differ. Equ. – 2006. – V.19. – 18 p.

2. Atik Y., Rakotoson J.-M. Local T-sets and degenerate variational problems. I// Appl. Math. Lett. - 1994. - V.7, №4. - P. 49-53.

3. Bendahmane M., Karlsen K.H. Nonlinear anisotropic elliptic and parabolic equations in RN with advection and lower order terms and locally integrable data// Potential Anal. - 2005. - V.22, №3. - P. 207-227.

4. Benilan Ph., Boccardo L., Gallouet T., Gariepy R., Pierre M., Vazquez J.L. An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations// Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). - 1995. - V.22, №2. - P. 241-273.

5. Boccardo L., Gallouet T. Nonlinear elliptic and parabolic equations involving measure data// J. Funct. Anal. - 1989. - V.87, №1. - P. 149-169.

6. Boccardo L., Gallouet T. Nonlinear elliptic equations with right hand side measures// Comm. Partial Differential Equations. - 1992. - V.17, №3-4. - P. 641-655.

7. Boccardo L., Gallouet T., Marcellini P. Anisotropic equations in $L^1$// Differential Integral Equations. - 1996. - V.9, №1. - P. 209-212.

8. Cavalheiro A.C. Existence of entropy solutions for degenerate quasilinear elliptic equations// Complex Var. Elliptic Equ. - 2008. - V.53, №10. - P. 945-956.

9. Cirmi G.R. On the existence of solutions to non-linear degenerate elliptic equations with measures data// Ricerche Mat. - 1993. - V.42, №2. - P. 315-329.

10. Kinderlehrer D., Stampacchia G. An introduction to variational inequalities and their applications. - M.: Mir, 1983. - 256 p. (in Russian)

11. Kovalevsky A.A., Gorban Yu.S. Degenerate anisotropic variational inequalities with $L^1$-data - Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Donetsk, 2007, preprint № 2007.01. - 92 p. (in Russian)

12. Kovalevsky A.A., Gorban Yu.S. Degenerate anisotropic variational inequalities with $L^1$-data// C. R. Math. Acad. Sci. Paris. - 2007. - V.345, №8. - P. 441-444.

13. Kovalevsky A.A., Gorban Yu.S. On T-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data// Izv. Math. - 2011. - V.75, №1. - P. 101-160. (in Russian)

14. Kovalevsky A.A., Gorban Yu.S. Solvability of degenerate anisotropic elliptic second-order equations with $L^1$-data// Electron. J. Differential Equations. - 2013. - №167. - P. 1-17.

15. Kovalevsky A.A. Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with $L^1$-right-hand sides// Izv. Math. - 2001. - V.65, №2. - P. 27-80. (in Russian)

16. Li F.Q. Nonlinear degenerate elliptic equations with measure data// Comment. Math. Univ. Carolin. - 2007. - V.48, №4. - P. 647-658.

	On uniqueness of entropy solutions for nonlinear elliptic degenerate anisotropic equations
Author	Yu. S. Gorban yuliya_gorban@ mail.ru Donetsk National University, Vinnytsia, Ukraine
Abstract	In the present paper we deal with the Dirichlet problem for a class of degenerate anisotropic elliptic second-order equations with $L^1$-right-hand sides in a bounded domain of ${\Bbb R}^n$ $(n \geqslant 2)$. This class is described by the presence of a set of exponents $q_1,\dots,q_n$ and a set of weighted functions $\nu_1,\dots,\nu_n$ in growth and coercitivity conditions on coefficients of the equations. The exponents $q_i$ characterize the rates of growth of the coefficients with respect to the corresponding derivatives of unknown function, and the functions $\nu_i$ characterize degeneration or singularity of the coefficients with respect to independent variables. Our aim is to study the uniqueness of entropy solution of the problem under consideration.
Keywords	nonlinear elliptic degenerate anisotropic second-order equations; $L^1$-data; Dirichlet problem; uniqueness of entropy solution
DOI	doi:10.15330/ms.47.1.59-70
Reference	1. Aharouch L., Azroul E., Benkirane A. Quasilinear degenerated equations with $L^1$ datum and without coercivity in perturbation terms// Electron. J. Qual. Theory Differ. Equ. – 2006. – V.19. – 18 p. 2. Atik Y., Rakotoson J.-M. Local T-sets and degenerate variational problems. I// Appl. Math. Lett. - 1994. - V.7, №4. - P. 49-53. 3. Bendahmane M., Karlsen K.H. Nonlinear anisotropic elliptic and parabolic equations in RN with advection and lower order terms and locally integrable data// Potential Anal. - 2005. - V.22, №3. - P. 207-227. 4. Benilan Ph., Boccardo L., Gallouet T., Gariepy R., Pierre M., Vazquez J.L. An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations// Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). - 1995. - V.22, №2. - P. 241-273. 5. Boccardo L., Gallouet T. Nonlinear elliptic and parabolic equations involving measure data// J. Funct. Anal. - 1989. - V.87, №1. - P. 149-169. 6. Boccardo L., Gallouet T. Nonlinear elliptic equations with right hand side measures// Comm. Partial Differential Equations. - 1992. - V.17, №3-4. - P. 641-655. 7. Boccardo L., Gallouet T., Marcellini P. Anisotropic equations in $L^1$// Differential Integral Equations. - 1996. - V.9, №1. - P. 209-212. 8. Cavalheiro A.C. Existence of entropy solutions for degenerate quasilinear elliptic equations// Complex Var. Elliptic Equ. - 2008. - V.53, №10. - P. 945-956. 9. Cirmi G.R. On the existence of solutions to non-linear degenerate elliptic equations with measures data// Ricerche Mat. - 1993. - V.42, №2. - P. 315-329. 10. Kinderlehrer D., Stampacchia G. An introduction to variational inequalities and their applications. - M.: Mir, 1983. - 256 p. (in Russian) 11. Kovalevsky A.A., Gorban Yu.S. Degenerate anisotropic variational inequalities with $L^1$-data - Institute of Applied Mathematics and Mechanics of NAS of Ukraine, Donetsk, 2007, preprint № 2007.01. - 92 p. (in Russian) 12. Kovalevsky A.A., Gorban Yu.S. Degenerate anisotropic variational inequalities with $L^1$-data// C. R. Math. Acad. Sci. Paris. - 2007. - V.345, №8. - P. 441-444. 13. Kovalevsky A.A., Gorban Yu.S. On T-solutions of degenerate anisotropic elliptic variational inequalities with $L^1$-data// Izv. Math. - 2011. - V.75, №1. - P. 101-160. (in Russian) 14. Kovalevsky A.A., Gorban Yu.S. Solvability of degenerate anisotropic elliptic second-order equations with $L^1$-data// Electron. J. Differential Equations. - 2013. - №167. - P. 1-17. 15. Kovalevsky A.A. Entropy solutions of Dirichlet problem for a class of nonlinear elliptic fourth order equations with $L^1$-right-hand sides// Izv. Math. - 2001. - V.65, №2. - P. 27-80. (in Russian) 16. Li F.Q. Nonlinear degenerate elliptic equations with measure data// Comment. Math. Univ. Carolin. - 2007. - V.48, №4. - P. 647-658.
Pages	59-70
Volume	47
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On uniqueness of entropy solutions for nonlinear elliptic degenerate anisotropic equations