Abstract
In this paper a class of N-dimensional second-order linear elliptic equations with a drift is studied. When the drift belongs to L 2 the existence of a renormalized solution is proved. There is also uniqueness in the class of the renormalized solutions modulo \(L^{\infty }\), but the uniqueness is violated when the drift equation is regarded in the distributions sense. Then, considering a sequence of oscillating drifts which converges weakly in L 2 to a limit drift in L q, with q > N, the homogenization process makes appear an extra zero-order term involving a non-negative Radon measure which does not load the zero capacity sets. This extends the homogenization result obtained in [3] by relaxing the equi-integrability of the drifts in L 2.
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Bénilan, P., Boccardo, L., Gallouët, 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, Serie IV 22(2), 241–273 (1995)
Briane M.: Homogenization with an oscillating drift: from L 2-bounded to unbounded drifts, 2d compactness results and 3d nonlocal effect. Ann. Mate. Pura Appl. 192(5), 853–878 (2013)
Briane, M, Gérard, P.: A drift homogenization problem revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(5), 1–39 (2012)
Casado-Díaz, J., Dal Maso, G.: A weak notion of convergence in capacity with applications to thin obstacle problems, Calculus of Variations and Related Topics. In: Ioffe, A., Reich, S., Shafrir, I. (eds.) in Research Notes in Mathematics 410, Chapman and Hall, Boca Raton, 56–64 (2000)
Dal Maso, G., Garroni, A.: New results on the asymptotic behavior of Dirichlet problems in perforated domains. Math. Mod. Meth. Appl. Sci. 3, 373–407 (1994)
Dal Maso, G., Murat, F.: Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 24(2), 239–290 (1997)
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(4), 741–808 (1999)
Geymonat, G.: G. Sul problema di Dirichlet per le equazioni lineari ellittiche. Ann. Scuola Norm. Sup. Pisa 16(3), 225–284 (1962)
Lions, P.-L., Murat, F.: Sur les solutions renormalisées d’quations elliptiques non linéaires, unpublished.
Meyers, N.G.: An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3(17), 189–206 (1963)
Murat, F.: Soluciones renormalizadas de EDP elipticas no lineales. Prépublication du laboratoire d’Analyse numérique de l’université Paris VI , France (1993)
Serrin, J.: Pathological solutions of elliptic differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 18(3), 385–387 (1964)
Stampacchia, G.: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier 15, 189–258 (1965)
Tartar, L.: Homogénéisation en hydrodynamique, in Singular Perturbation and Boundary Layer Theory, Lecture Notes Math., 597, Springer, Berlin-Heidelberg, 474–481 (1977)
Tartar, L.: Remarks on homogenization, in Homogenization and effective moduli of materials and media, IMA Vol. Math. Appl., 1, Springer, New-York, 228–246 (1986)
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Briane, M., Casado-Díaz, J. A Class of Second-Order Linear Elliptic Equations with Drift: Renormalized Solutions, Uniqueness and Homogenization. Potential Anal 43, 399–413 (2015). https://doi.org/10.1007/s11118-015-9478-1
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DOI: https://doi.org/10.1007/s11118-015-9478-1