an:05949054
Zbl 1229.35010
Kim, Sunghoon; Lee, Ki-Ahm
Viscosity method for homogenization of parabolic nonlinear equations in perforated domains
EN
J. Differ. Equations 251, No. 8, 2296-2326 (2011).
00285958
2011
j
35B27 35K55 35K65 35D40 35K85 35K20
obstacle problem; corrector; porous medium equation; approximate penalized problem
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.
Alain Brillard (Riedisheim)