Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure

• 15 Accesses

• 2 Citations

Abstract

In this paper we deal with asymptotic behaviour of renormalized solutions $$u_{n}$$ to the nonlinear parabolic problems whose model is

\begin{aligned} {\left\{ \begin{array}{ll} (u_{n})_{t}-\text {div}(a_{n}(t,x,\nabla u_{n}))=\mu _{n}&{}\text { in }Q=(0,T)\times \Omega ,\\ u_{n}(t,x)=0&{}\text { on }(0,T)\times \partial \Omega ,\\ u_{n}(0,x)=u_{0}^{n}&{}\text { in }\Omega , \end{array}\right. } \end{aligned}

where $$\Omega$$ is a bounded open set of $$\mathbb {R}^{N}$$, $$N\ge 1$$, $$T>0$$ and $$u_{0}^{n}\in C^{\infty }_{0}(\Omega )$$ that approaches $$u_{0}$$ in $$L^{1}(\Omega )$$. Moreover $$(\mu _{n})_{n\in \mathbb {N}}$$ is a sequence of Radon measures with bounded variation in Q which converges to $$\mu$$ in the narrow topology of measures. The main result states that, under the assumption of G-convergence of the operators $$A_{n}(v)=-\text {div}(a_{n}(t,x,\nabla v_{n}))$$, defined for $$v_{n}\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))$$ for $$p>1$$, to the operator $$A_{0}(v)=-\text {div}(a_{0}(t,x,\nabla v))$$ and up to subsequences, $$(u_{n})$$ converges a.e. in Q to the renormalized solution u of the problem

\begin{aligned} {\left\{ \begin{array}{ll} u_{t}-\text {div}(a_{0}(t,x,\nabla u))=\mu &{}\text { in }Q=(0,T)\times \Omega ,\\ u(t,x)=0&{}\text { on }(0,T)\times \partial \Omega ,\\ u(0,x)=u_{0}&{}\text { in }\Omega . \end{array}\right. } \end{aligned}

The proposed renormalized formulation differs from the usual one by the fact that truncated function $$T_{k}(u_{n})$$ (which depend on the solutions) are used in place of the solutions $$u_{n}$$. We prove existence of such a limit-solution and we discuss its main properties in connection with G-convergence, we finally show the relationship between the new approach and the previous ones and we extend this result using capacitary estimates and auxiliary test functions.

This is a preview of subscription content, log in to check access.

References

1. 1.

Abdellaoui, M., Azroul, E.: Renormalized solutions for nonlinear parabolic equations with general measure data. Electron. J. Differ. Equ. 132, 1–21 (2018)

2. 2.

Abdellaoui, M., Azroul, E.: Renormalized solutions to nonlinear parabolic problems with blowing up coefficients and general measure data. Ric. Mat. 2019, 1–23 (2019)

3. 3.

Ben Cheikh Ali, M.: Homogénéisation des solutions renormalisées dans des domaines perforés. Thèse, Université de Rouen (2001)

4. 4.

Bénilan, P., Boccardo, L., Gallouët, T., Gariepy, R., Pierre, M., Vázquez, J.L.: An $$L^{1}-$$theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22, 241–273 (1995)

5. 5.

Bensoussan, A., Boccardo, L., Murat, F.: $$H-$$convergence for quasilinear elliptic equations with quadratic growth. Appl. Math. Optim. 26, 253–272 (1992)

6. 6.

Boccardo, L.: Homogeneisation pour une classe d’equations fortement nonlineaires. C. R. Acad. Sci. Paris 306, 253–256 (1988)

7. 7.

Boccardo, L.: Homogenization and continuous dependence for Dirichlet problems in $$L^{1}$$. Partial Differential Equation Methods in Control and Shape Analysis (Pisa 1994), Lecture Notes in Pure and Applied Mathematics, vol. 188, pp. 41–52. Dekker, New York (1997)

8. 8.

Boccardo, L., Gallouët, T.: Homogenization with jumping nonlinearities. Ann. Mat. Pura Appl. (4) 138, 211–221 (1984)

9. 9.

Boccardo, L., Gallouët, T.: Nonlinear elliptic equations with right-hand side measures. Comm. Partial Differ. Equ. 17(3&3), 641–655 (1992)

10. 10.

Boccardo, L., Murat, F.: Remarques sur l’homogeneisation de certaines problemes quasilineaires. Portugal. Math. 41, 535–562 (1982)

11. 11.

Boccardo, L., Orsina, L.: Existence results for Dirichlet Problems in $$L^{1}$$ via Minty’s Lemma. Appl. Anal. 76, 309–317 (2000)

12. 12.

Boccardo, L., Del Vecchio, T.: Homogenization of strongly nonlinear equations with gradient dependent lower order nonlinearity. Asymptot. Anal. 5, 75–90 (1991)

13. 13.

Bögelein, V., Duzaar, F., Mingione, G.: Degenerate problems with irregular obstacles. J. Reine Angew. Math. 650, 107–160 (2011)

14. 14.

Braides, A.: $$\Gamma$$-Convergence for Beginners, Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002)

15. 15.

Briane, M., Casado-Diáz, J.: A class of second-order linear elliptic equations with drift: renormalized solutions, uniqueness and homogenization. Potential Anal. 43(3), 399–413 (2015)

16. 16.

Casado Diaz, J., Dal Maso, G.: A weak notion of convergence in capacity with applications to thin obstacle problems. In: Casado Diaz, J. (ed.) Calculus of Variations and Differential Equations (Haifa, 1998). Chapman and Hall, London (1999)

17. 17.

Casado-Diaz, J., Rebollo, T.C., Girault, V., Marmol, M.G., Murat, F.: Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in $$L^{1}$$. Numer. Math. 105, 337–374 (2007)

18. 18.

Chiado Piat, V., Dal Maso, G., Defranceschi, A.: $$G-$$convergence of monotone operators. Ann. Inst. H. Poincare. Anal. Non Lineaire 7, 123–160 (1990)

19. 19.

Colombini, F., Spagnolo, S.: Sur la convergence des solutions d’equations paraboliques avec des coefficients qui dependent du temps. C. R. Acad. Sci. Paris Ser. A 282, 735–737 (1976)

20. 20.

Colombini, F., Spagnolo, S.: Sur la convergence des solutions d’equations paraboliques. J. Math. Pures Appl. (9) 56, 263–306 (1977)

21. 21.

Colombini, F., Spagnolo, S.: On the convergence of solutions of hyperbolic equations. Commun. Partial Differ. Equ. 3, 77–103 (1978)

22. 22.

Dal Maso, G.: An Introduction to $$\Gamma$$-Convergence, Vol. 8 of Progress in Nonlinear Differential Equations and Their Applications. Birkhäuser, Boston (1993)

23. 23.

Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 28, 741–808 (1999)

24. 24.

Dall’Aglio, A.: Approximated solutions of equations with $$L^{1}$$ data. Application to the $$H$$-convergence of quasi-linear parabolic equations. Ann. Mat. Pura Appl. 170, 207–240 (1996)

25. 25.

De Arcangelis, R., Cassano, F.S.: On the convergence of solutions of degenerate elliptic equations in divergence form. Ann. Mat. Pura Appl. (IV) CLXVII, 1–23 (1994)

26. 26.

Defranceschi, A.: $$G-$$convergence of cyclically monotone operators. Asymptot. Anal. 2, 21–37 (1989)

27. 27.

Defranceschi, A.: Asymptotic analysis of boundary value problems for quasi-linear monotone operators. Asymptot. Anal. 3, 221–247 (1990)

28. 28.

Del Vecchio, T.: On the homogenization of a class of pseudomonotone operators in divergence form. Boll. Un. Mat. Ital. (7) 5–B, 369–388 (1991)

29. 29.

DiBenedetto, E.: Partial Differential Equations. Birkhäuser, Boston (1995)

30. 30.

Donato, P., Guibé, O., Oropeza, A.: Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and $$L^{1}$$ data. J. Mat. Pures Appl. 120, 91–129 (2018)

31. 31.

Droniou, J., Porretta, A., Prignet, A.: Parabolic capacity and soft measures for nonlinear equations. Potential Anal. 19(2), 99–161 (2003)

32. 32.

Franco, J.: Homogenization and correctors for nonlinear elliptic equations. Comment. Math. Univ. Carolin. In: Proceedings of the 7th Czechoslovak Conference on Differential Equations and Their Applications held in Prague (1989), pp. 167–170. BSB B.G. Teubner Verlagsgesellschaft, Leipzig, 1990

33. 33.

Fukushima, M., Sato, K., Taniguchi, S.: On the closable part of pre-Dirichlet forms and the fine supports of underlying measures. Osaka J. Math. 28, 517–535 (1991)

34. 34.

Gaudiello, A., Guibé, O., Murat, F.: Homogenization of the brush problem with a source term in $$L^{1}$$. Arch. Ration. Mech. Anal. 225(1), 1–64 (2017)

35. 35.

Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1983)

36. 36.

Giusti, E.: Metodi diretti nel Calcolo delle Variazioni. UMI, Bologna (1994)

37. 37.

Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Oxford University Press, Oxford (1993)

38. 38.

Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (2012)

39. 39.

Kamynin, V.L.: Convergence estimates for the solutions of certain quasi-linear parabolic equations with weakly convergent coefficients (in Russian). Sibirsk. Mat. Z. (Translated as: Siberian Math. J.) 29, 118–130 (1988)

40. 40.

Kamynin, V.L.: On passage to the limit in quasilinear elliptic equations with several independent variables. Math. USSR-Sb. 60, 47–66 (1988)

41. 41.

Kozlov, S.M.: Averaging of difference schemes. Math. USSR-Sb. 57, 351–369 (1987)

42. 42.

Kruzhkov, S.N., Kamynin, V.L.: Convergence of the solutions of quasilinear parabolic equations with weakly converging coefficients. Soviet Math. Dokl. 27, 533–536 (1983)

43. 43.

Kruzhkov, S.N., Kamynin, V.L.: On passage to the limit in quasilinear parabolic equations. Proc. Steklov Inst. Math. 1986, 205–232 (1986)

44. 44.

Landes, R., Mustonen, V.: A strongly nonlinear parabolic initial-boundary value problem. Ark. Mat. 25(1), 29–40 (1987)

45. 45.

Leray, J., Lions, J.-L.: Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty et Browder. Bull. Soc. Math. France 93, 97–107 (1965)

46. 46.

Lions, J.-L.: Quelques Méthodes de Résolution des Problèmes Aux Limites Non Linéaire. Dunod et Gautier-Villars, Paris (1969)

47. 47.

Mahadevan, R., Muthukumar, T.: Homogenization of some cheap control problems. SIAM J. Math. Anal. 43(5), 2211–2229 (2011)

48. 48.

Malusa, A., Orsina, L.: Asymptotic behaviour of renormalized solutions to elliptic equations with measure data and $$G-$$converging operators. Calc. Var. Partial Differ. Equ. 27(2), 179–202 (2006)

49. 49.

Malusa, A., Prignet, A.: Stability of renormalized solutions of elliptic equations with measure data. Atti Sem. Mat. Fis. Univ. Modena 52, 117–134 (2004)

50. 50.

Murat, F.: $$H-$$convergence. Seminaire d’Analyse Fonctionnelle et Numerique de l’Universite d’Alger (1977)

51. 51.

Murat, F.: Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 489–507 (1978)

52. 52.

Murat, F.: Homogenization of renormalized solutions of elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 8, 309–332 (1991)

53. 53.

Murat, F., Tartar, L.: $$H-$$convergence. In: Murat, F. (ed.) Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 21–43. Birkhäuser, Boston (1997)

54. 54.

Ngoan, K.T.: On the convergence of solutions of boundary-value problems for sequences of elliptic systems. Moscow Univ. Math. Bull. 32, 66–74 (1977)

55. 55.

Oleinik, O.A.: On the convergence of solutions of elliptic and parabolic equations under weak convergence of coefficients. Uspehi Mat. Nauk (4) 30, 257–258 (1975). (transl. as Russian Math. Surveys)

56. 56.

Pankov, A.A.: $$G-$$convergence and homogenization for nonlinear elliptic operators (in Russian). Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 2–88 (1988)

57. 57.

Pankov, A.A.: $$G$$-Convergence and Homogenization of Nonlinear Partial Differential Operators. Vinnitsa Polytechnical Institute, Vinnitsa, Ukraine

58. 58.

Pankov, A.A.: Averaging of elliptic operators with strong nonlinearity in lower order terms (in Russian). Differ. Uravn. 23, 1786–1791 (1987)

59. 59.

Petitta, F.: Nonlinear parabolic equations with general measure data. Ph.D. Thesis. Università di Roma, Italy (2006)

60. 60.

Petitta, F.: Asymptotic behavior of solutions for linear parabolic equations with general measure data. C. R. Acad. Sci. Paris Ser. I344, 571–576 (2007)

61. 61.

Petitta, F.: Renormalized solutions of nonlinear parabolic equations with general measure data. Ann. Mat. Pura Appl. 187(4), 563–604 (2008)

62. 62.

Petitta, F.: A non-existence result for nonlinear parabolic equations with singular measure data. Proc. R. Soc. Edinb. Sect. A Math. 139, 381–392 (2009)

63. 63.

Petitta, F., Porretta, A.: On the notion of renormalized solution to nonlinear parabolic equations with general measure data. J. Ellipt. Parabol. Equ. 1, 201–214 (2015)

64. 64.

Petitta, F., Ponce, A.C., Porretta, A.: Diffuse measures and nonlinear parabolic equations. J. Evol. Equ. 11(4), 861–905 (2011)

65. 65.

Pierre, M.: Parabolic capacity and Sobolev spaces. Siam J. Math. Anal. 14, 522–533 (1983)

66. 66.

Prignet, A.: Remarks on existence and uniqueness of solutions of elliptic problems with right hand side measures. Rend. Mat. 15, 321–337 (1995)

67. 67.

Prignet, A.: Existence and uniqueness of entropy solutions of parabolic problems with $$L^{1}$$ data. Nonlin. Anal. TMA 28, 1943–1954 (1997)

68. 68.

Sbordone, C.: Alcune questioni di convergenza per operatori differenziali del secondo ordine. Boll. Un. Mat. Ital. (4) 10, 672–682 (1974)

69. 69.

Senatorov, P.K.: The stability of the solution of a Dirichlet problem for an elliptic equation under perturbations (in measure) of its coefficients. Differ. Equ. 6, 1312–1313 (1970)

70. 70.

Simon, L.M.: On $$G-$$convergence of elliptic operators. Indiana Univ. Math. J. 28, 587–594 (1979)

71. 71.

Spagnolo, S.: Convergence in energy for elliptic operators. In: Proceedings of Third Symposium on Numerical Solution of Partial Differential Equations (College Park, 1975), pp. 469–498. Academic Press, New York (1976)

72. 72.

Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22, 577–597 (1968)

73. 73.

Spagnolo, S.: Sui limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 21, 657–699 (1967)

74. 74.

Spagnolo, S.: Convergence of parabolic equations. Boll. Un. Mat. Ital. (5) 14–B, 547–568 (1977)

75. 75.

Spagnolo, S.: Perturbation des Coefficients d’une Equation Hyperbolique du Deuxieme Ordre. Nonlinear Partial Differential Equations and Their Applications. College de Prance Seminar. Vol. 1, 365–377 (1978–1979), Res. Notes in Math. 53. Pitman, London (1980)

76. 76.

Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du seconde ordre à coefficientes discontinus. Ann. Inst. Fourier (Grenoble) 15, 189–258 (1965)

77. 77.

Tartar, L.: Compensated compactness and applications to partial differential equations. In: Nonlinear Analysis and Mechanics. Heriot–Watt Symposium, vol. IV, Research Notes in Mathematics, 39. Pitman, London (1979)

78. 78.

Tartar, L.: Quelques remarques sur l’homogeneisation. In: Proceedings of the Japan–France Seminar 1976 “Functional Analysis and Numerical Analysis”, pp. 469–482. Japan Society for the Promotion of Science (1978)

79. 79.

Zhikov, V.V.: $$G-$$convergence of elliptic operators (in Russian). Mat. Zametki 33, 345–356 (1983)

80. 80.

Zhikov, V.V.: Asymptotic behaviour and stabilization of solutions of parabolic equations with lower terms. Trans. Moscow Math. Soc. 46, 66–99 (1984)

81. 81.

Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: $$G-$$convergence of parabolic operators. Russ. Math. Surv. 36, 9–60 (1981)

Author information

Correspondence to Mohammed Abdellaoui.

Ethics declarations

Conflict of interest

The authors would like to thank Pr. Alexander Pankov, the reviewers for their thoughtful comments towards improving our manuscript. On behalf of all authors, the corresponding author states that there is no conflict of interest.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

Abdellaoui, M., Azroul, E. Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure. SeMA 77, 1–26 (2020). https://doi.org/10.1007/s40324-019-00197-8

• Accepted:

• Published:

• Issue Date:

Keywords

• Nonlinear parabolic problems
• Homogenization (G-convergence)
• Measure data

• 35R06
• 32U20
• 80M40