The present paper is devoted to study the asymptotic behavior of a sequence of linear elliptic equations with a varying drift term, whose coefficients are just bounded in , with N the dimension of the space. It is known that there exists a unique solution for each of these problems in the Sobolev space . However, because the operators are not coercive, there is no uniform estimate of the solutions in this space. We use some estimates in (J. Differential Equations258 (2015) 2290–2314), and a regularization obtained by adding a small nonlinear first order term, to pass to the limit in these problems.
P.Bénilan, L.Boccardo, T.Gallouët, R.Gariepy, R.Pierre and J.L.Vazquez, An theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa, Serie IV22 (1995), 241–273.
2.
L.Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations258 (2015), 2290–2314. doi:10.1016/j.jde.2014.12.009.
3.
L.Boccardo, Stampacchia–Calderón–Zygmund theory for linear elliptic equations with discontinuous coefficients and singular drift, ESAIM Control Optim. Calc. Var.25 (2019), 47, 13 pp.
4.
L.Boccardo, Weak maximum principle for Dirichlet problems with convection or drift terms, Math. Eng.3 (2021), 026, 9 pp.
5.
L.Boccardo, The impact of the zero order term in the study of Dirichlet problems with convection or drift terms, Rev. Mat. Complut.36 (2023), 571–605. doi:10.1007/s13163-022-00434-1.
6.
L.Boccardo and J.Casado-Díaz, H-convergence of singular solutions of some Dirichlet problems with terms of order one, Asymptot. Anal.64 (2009), 239–249.
7.
L.Boccardo, J.Casado-Díaz and L.Orsina, Dirichlet problems with skew-symmetric drift-terms, C. R. Math. Acad. Sci. Paris362 (2024), 301–306.doi:10.1016/0022-1236(89)90005-0
8.
L.Boccardo and T.Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal.87 (1989), 149–169. doi:10.1016/0022-1236(89)90005-0.
9.
L.Boccardo and F.Murat, Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations, Nonlinear Anal.19 (1992), 581–597. doi:10.1016/0362-546X(92)90023-8.
10.
M.Briane and J.Casado-Díaz, A class of second-order linear elliptic equations with drift: Renormalized solutions, uniqueness and homogenization, Potential Anal.43 (2015), 399–413. doi:10.1007/s11118-015-9478-1.
11.
M.Briane and J.Casado-Díaz, Homogenization of an elastodynamic system with a strong magnetic field and soft inclusions inducing a viscoelastic effective behavior, J. Math. Anal. Appl.492 (2020), 124472, 24 pp.
12.
M.Briane and J.Casado-Díaz, Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems, Calc. Var. and PDE60 (2021), 163, 39 pp.
13.
M.Briane and P.Gérard, A drift homogenization problem revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci.11 (2012), 1–39.
14.
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.28 (1999), 741–808.
15.
D.Gilbarg and N.Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 1998.
16.
L.Tartar, Homogénéisation en hydrodynamique, in: Singular Perturbation and Boundary Layer Theory, Lecture Notes Math., Vol. 597, Springer, Berlin–Heidelberg, 1977, pp. 474–481. doi:10.1007/BFb0086103.
