On some degenerate singular perturbation problems. (English) Zbl 0988.35061

Let \(\Omega\subset\mathbb{R}^n(n\geq 1)\) be a smooth bounded domain and \(\Omega'\subset\Omega\) a set of positive measure. The authors consider the following boundary value problem with a small parameter \(\varepsilon>0:\) \[ -\varepsilon \text{div}\bigl(a(x, u_\varepsilon)\nabla u_\varepsilon \bigr)+a_0 \chi_{\Omega'}u_\varepsilon=f\text{ in }\Omega,\;u_\varepsilon=0\text{ on } \partial \Omega, \] where a bounded Carathéodory matrix function \(a(x,u)\), \(x\in \Omega\), \(u\in\mathbb{R}\), satisfies the uniform ellipticity condition in \(\Omega\) (uniformly in \(\mathbb{R})\) and the Lipschitz condition in \(\mathbb{R}\) (uniformly in \(\Omega)\); \(a_0\) is a positive bounded function on \(\Omega\). The problem has a unique solution \(u_\varepsilon\in H^1(\Omega)\) for each \(f\in H^{-1} (\Omega)\). The main results on the asymptotic behavior of solutions \(u_\varepsilon\) (as \(\varepsilon\to 0)\) are as follows:
1. \(\varepsilon u_\varepsilon\to u_0\) in \(H^1(\Omega)\), where \(u_0\) is a solution to the problem \(\text{div} a(x,u_0) \nabla u_0=f\) in \(\Omega\), \(u_0=0\) in \(\Omega'\).
2. If \(\Omega'\) is a smooth open set, then \(a_0u_\varepsilon\to f\) in \(D'(\Omega')\); if \(f\in L^2_{\text{loc}} (\Omega')\), then \(u_\varepsilon\to f/a_0\) in \(L^2_{\text{loc}} (\Omega')\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35B25 Singular perturbations in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs