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Homogenization of monotone operators in divergence form with \(x\)-dependent multivalued graphs. (English) Zbl 1180.35077

The authors prove an homogenization result, in the sense of the H-convergence, for steady equations involving monotone graphs. The original problem is written as \(-\text{div}(d^{\varepsilon })=f\) in a smooth and bounded domain \(\Omega \) of \(\mathbb{R}^{N}\), with homogeneous Dirichlet boundary conditions on \(\partial \Omega \). Here \(f\) belongs to \(W^{-1,p'}(\Omega )\) and \(\text{grad}u^{\varepsilon }\) and \( d^{\varepsilon}\) respectively belonging to \(L^{p}(\Omega ;\mathbb{R}^{N})\) and \(L^{p'}(\Omega ;\mathbb{R}^{N})\) are linked through \( d^{\varepsilon }(x)-\text{grad}u^{\varepsilon }(x)=\varphi ^{\varepsilon}(x,d^{\varepsilon }(x)+\text{grad}u^{\varepsilon }(x))\) for some function \( \varphi ^{\varepsilon }\) which belongs to a set \(M^{\ast}(\alpha,m,m,\Omega)\) of functions.
The main result of the paper proves the convergence of \((u^{\varepsilon},d^{\varepsilon})\), up to some subsequence, to a solution \((u,d)\) of a similar problem but involving a limit function \(\varphi^{0}\) which belongs to the set \(M^{\ast}(\alpha ,m,m,\Omega)\). The paper starts proving an existence result for such problems. The main part of the paper is devoted to the convergence result. Notice that the dependence of \(\varphi^{\varepsilon}\) on \(\varepsilon\) is not necessarily obtained through a periodic process. The authors also compare their convergence result with that obtained by V. Chiad ò-Piat, G. Dal Maso and A. Defranceschi in [Ann. Inst. Henri Poincaré , Anal. Non Linéaire 7, No. 3, 123–160 (1990; Zbl 0731.35033)], using the G-convergence.
The paper ends with some results concerning maximal extensions of monotone graphs which extend the well-known results from [V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Leyden, The Netherlands: Noordhoff International Publishing (1976; Zbl 0328.47035)] and [H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies Vol. 5. Amsterdam-London: North-Holland Publishing Comp.; New York: American Elsevier Publishing Comp., Inc. (1973; Zbl 0252.47055)].

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
47H05 Monotone operators and generalizations
35J25 Boundary value problems for second-order elliptic equations
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[1] Barbu, V., Nonlinear semigroups and differential equations in Banach spaces (1976), Leiden: Noordhoff International, Leiden · Zbl 0328.47035
[2] Brezis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, vol. 5, Notas de Matemática (50). North-Holland, Amsterdam; American Elsevier, New York (1973) · Zbl 0252.47055
[3] Chiadò Piat, V.; Dal Maso, G.; Defranceschi, A., G-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7, 3, 123-160 (1990) · Zbl 0731.35033
[4] Francfort, G.; Murat, F., Tartar, L., Monotone operators in divergence form with x-dependent multivalued graphs, Boll. Un. Mat. Ital., 7, 23-59 (2004) · Zbl 1115.35047
[5] Murat, F., Tartar, L.: H-convergence. In: Cherkaev, A., Kohn, R.V. (eds.) Topics in the mathematical modeling of composite materials. Progress in Nonlinear differential equations and their applications, vol. 31, pp. 21-43. Birkhäuser, Basel (1997). (English translation of Murat, F., H-convergence, Séminaire d’analyse fonctionnelle et numérique, Université d’Alger, 1977-1978) · Zbl 0920.35019
[6] Tartar, L.: An introduction to the homogenization method in optimal design. In: Cellina, A., Ornelas, A. (eds) Optimal Shape Design, Lectures given at the joint C.I.M./C.I.M.E. summer school held in Tróia, June 1-6, 1998. Lecture Notes in Mathematics, vol. 1740, pp. 47-156. Springer, Berlin (2000) · Zbl 1040.49022
[7] Tartar, L.: The general theory of homogenization (to appear) · Zbl 1188.35004
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