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Viscosity method for homogenization of parabolic nonlinear equations in perforated domains. (English) Zbl 1229.35010
The main purpose of the paper is to establish homogenization results for parabolic nonlinear equations in perforated domains. The authors start with the obstacle problem \(\Delta u_\varepsilon-u_t\leq 0\) in \(\Omega \times (0,T]\), with \(u_\varepsilon\geq \varphi_\varepsilon\) in \(\Omega \times (0,T] \), the solution starting from an initial data \(g\) at \(t=0\) and satisfying homogeneous Dirichlet boundary conditions on \(\partial \Omega \times (0,T]\). Here \(\Omega \) is a smooth, bounded and connected domain of \(\mathbb R^n\) and \(\varphi_\varepsilon\) is taken as \(\varphi \chi_{{\mathcal T} _{a_\varepsilon}}\) where \(\varphi \) is a smooth function which is negative on the boundary \(\partial \Omega \times (0,T]\), and \({\mathcal T}_{a_\varepsilon}\) is the union of \(\varepsilon\)-cells from which have been removed the spherical balls centered at the center of the cells and of radius \( a_\varepsilon\).
The first main result of the paper describes the asymptotic behaviour of the least viscosity super-solution of this obstacle problem. The authors distinguish between three cases according to the decay rates of \( a_\varepsilon\). The main step of the proof consists of introducing the solution \(u_{\varepsilon,\delta }\) of an approximate penalized problem obtained when introducing in the previous equation the penalized term \(\beta _\delta(u_{\varepsilon,\delta }(x,t)-\varphi_\varepsilon(x,t))\), where \( \beta _\delta\) is some penalty function. The last part of the paper deals with a porous medium \(\Delta u_\varepsilon^m-\partial _t u_\varepsilon=0\) posed in the perforated domain with homogeneous Dirichlet boundary conditions on \(\partial \Omega \times (0,T]\). Here \(m\in (1,\infty )\) and the solution starts from an initial data \(g_\varepsilon=g\xi \) where \( g\in C_0^\infty(\Omega )\) and \(\xi \in C^\infty\) is an \(\varepsilon\)-periodic function which is the solution of a Laplace equation in a perforated domain. For this porous medium equation, the authors perform the transformation \(v_\varepsilon=u_\varepsilon^m\) and establish the equation satisfied by \(v_\varepsilon\). The main result here builds the limit of \( v_\varepsilon\) when \(\varepsilon\) goes to 0.
MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35D40 Viscosity solutions to PDEs
35K85 Unilateral problems for linear parabolic equations and variational inequalities with linear parabolic operators
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] Allaire, Grégoire, Homogenization and two-scale convergence, SIAM J. math. anal., 23, 6, 1482-1518, (1992) · Zbl 0770.35005
[2] Attouch, Hédy; Picard, Colette, Variational inequalities with varying obstacles: the general form of the limit problem, J. funct. anal., 50, 2, 329-386, (1983)
[3] Baffico, L.; Conca, C.; Rajesh, M., Homogenization of a class of nonlinear eigenvalue problems, Proc. roy. soc. Edinburgh sect. A, 136, 1, 7-22, (2006) · Zbl 1105.35010
[4] Bensoussan, Alain; Lions, Jacques-Louis; Papanicolaou, George, Asymptotic analysis for periodic structures, Stud. math. appl., ISBN: 0-444-85172-0, vol. 5, (1978), North-Holland Publishing Co. Amsterdam, New York, xxiv+700 pp · Zbl 0404.35001
[5] Caffarelli, L.A., A note on nonlinear homogenization, Comm. pure appl. math., 52, 7, 829-838, (1999) · Zbl 0933.35022
[6] Carbone, L.; Colombini, F., On convergence of functionals with unilateral constraints, J. funct. anal., 15, 329-386, (1983)
[7] Caffarelli, Luis; Lee, Ki-ahm, Viscosity method for homogenization of highly oscillating obstacles, Indiana univ. math. J., 57, 1715-1742, (2008) · Zbl 1158.35109
[8] Caffarelli, L.; Lee, Ki-Ahm, Homogenization of nonvariational viscosity solutions, Rend. accad. naz. sci. XL mem. mat. appl. (5), 29, 1, 89-100, (2005)
[9] Caffarelli, L.; Lee, K., Homogenization of oscillating free boundaries: the elliptic case, Comm. partial differential equations, 32, 1-3, 149-162, (2007) · Zbl 1122.35156
[10] Caffarelli, Luis A.; Lee, Ki-Ahm; Mellet, Antoine, Singular limit and homogenization for flame propagation in periodic excitable media, Arch. ration. mech. anal., 172, 2, 153-190, (2004) · Zbl 1058.76070
[11] Caffarelli, L.A.; Lee, K.-A.; Mellet, A., Homogenization and flame propagation in periodic excitable media: the asymptotic speed of propagation, Comm. pure appl. math., 59, 4, 501-525, (2006) · Zbl 1093.35010
[12] Caffarelli, L.A.; Lee, K.-A.; Mellet, A., Flame propagation in one-dimensional stationary ergodic media, Math. models methods appl. sci., 17, 1, 155-169, (2007) · Zbl 1110.76054
[13] Cioranescu, Doina; Murat, Francois, A strange term coming from nowhere, (), 45-93, (english version) · Zbl 0912.35020
[14] Cioranescu, Doina; Murat, Francois, Un terme étrange venu dʼailleurs I, (), 98-138, (french version) · Zbl 0498.35034
[15] Cioranescu, Doina; Murat, Francois, Un terme étrange venu dʼailleurs II, (), 154-178, (french version) · Zbl 0498.35034
[16] Caffarelli, Luis A.; Souganidis, Panagiotis E.; Wang, L., Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. pure appl. math., 58, (2005) · Zbl 1063.35025
[17] Dal Maso, G., Limiti di soluzioni di problemi varizionali con ostacoli bilaterali, Atti. accad. naz. lincei rend. cl. sci. fis. mat. natur., 69, 333-337, (1980) · Zbl 0498.49009
[18] Dal Maso, G., Asymptotic behavior of minimum problems with bilateral obstacles, Ann. mat. pura appl., 129, 327-366, (1981)
[19] De Giorgi, Ennio, G-operators and γ-convergence, (), 1175-1191
[20] Maso, G. Dal; Longo, P., γ-limits of obstacles, Ann. mat. pura appl., 128, 1-50, (1981) · Zbl 0467.49004
[21] De Diorgi, E.; Maso, G. Dal; Longo, P., γ-limiti di ostacoli, Atti. acad. naz. lincei rend. cl. sci. fis. mat. natur.,68., 481-487, (1980)
[22] Evans, L.C., Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. roy. soc. Edinburgh sect. A, 120, 3-4, 245-265, (1992) · Zbl 0796.35011
[23] Friedman, A., Variational principles and free-boundary problems, (1988), Robert E. Krieger Publishing Co. · Zbl 0671.49001
[24] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehren math. wiss., vol. 224, (1977), Springer-Verlag Heidelberg · Zbl 0691.35001
[25] Jikov, V.V.; Kozlov, S.M.; Oleinik, O.A., Homogenization of differential operators and integral functionals, ISBN: 3-540-54809-2, (1994), Springer-Verlag Berlin, xii+570 pp., translated from the Russian by G.A. Yosifian [G.A. Iosif yan] · Zbl 0801.35001
[26] Ki-Ahm Lee, Obstacle problem for nonlinear \(2^{n d}\)-order elliptic operator, PhD thesis, New York University, 1998.
[27] Lee, Ki-Ahm, The obstacle problem for Monge-Ampére equation, Comm. partial differential equations, 26, 1-2, 33-42, (2001) · Zbl 0982.35039
[28] Lieberman, G.M., Second order parabolic partial differential equations, (1996), World Scientific · Zbl 0884.35001
[29] Nguetseng, Gabriel, A general convergence result for a functional related to the theory of homogenization, SIAM J. math. anal., 20, 3, (1989) · Zbl 0688.35007
[30] Nandakumaran, A.K.; Rajesh, M., Homogenization of a parabolic equation in perforated domain with Dirichlet boundary condition, Proc. Indian acad. sci. math. sci., 112, 3, 425-439, (2002) · Zbl 1021.35009
[31] Spagnolo, Sergio, Convergence in energy for elliptic operators, (), 469-498 · Zbl 0347.65034
[32] Tartar, L., Compensated compactness and applications to partial differential equations, (), 136-212 · Zbl 0437.35004
[33] Tartar, Luc, Topics in nonlinear analysis, Publ. math. dʼorsay, vol. 7813, (1978), University de Paris-Sud, Departement de Mathematique Orsay, ii+271 pp · Zbl 0395.00008
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