## 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 $$L^2_{\text{loc}} (\Omega')$$.

### MSC:

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