Let $\Omega$ be a open, bounded domain in $R^n$. Let $Lu=\sum (a_{ij}(x)u_{x_i})_{x_j} +c(x) u$ be the operator where $a_{ij}=a_{ji}$ are in $C^1$ and $c(x)$ is continuous. Let $\Omega_0$ be a bounded open set with $C^2$ boundary, whose closure is compact in $\Omega$. Let also $\Omega_\epsilon=\{x\in\Omega : dist(x,\Omega_0)<\epsilon\}$. We need to show that $\lambda(\Omega_\epsilon)\to\lambda(\Omega_0)$ as $\epsilon\to0$. Here $\lambda(\Omega_\epsilon)$ is the first Dirichlet Eigen value of $L$ in $\Omega_\epsilon$ and $L$ is uniformly elliptic operator.
I tried with relating the first Eigen value with the Rayleigh quotient and then using the Poincare inequality but I did not succeeded to proving anything. Since the minimizer changes with domain changing I have no idea how to approach. Any help is greatly appreciated.
The result is indeed true, and it is valid under very general settings. As was pointed out in the comments by MaoWao, the convergence takes place provided $H^1(\Omega) = \cap_{\varepsilon>0} H_0^1(\Omega_\varepsilon)$. And this equality is known to be true if, in particular, $\Omega$ is of class $C^2$.
See [Babuška, I., & Výborný, R. (1965). Continuous dependence of eigenvalues on the domain. Czechoslovak Mathematical Journal, 15(2), 169-178] and [Fuglede, B. (1999). Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space. Journal of Functional Analysis, 167(1), 183-200], for independent expositions about the convergence.