# Eigenfunction associated to the smallest eigenvalue of an elliptic operator

Let $(T_n)$ be a sequence of elliptic operators defined in $H^2(\Omega)\cap H_0^1(\Omega)$ to $L^2(\Omega)$, with $\Omega$ being a bounded domain with smooth boundary. All of them have a smallest eigenvalue $\lambda_n$ to which we associate a positive eigenfunction $\varphi_n$, with norm equal to 1.

Now, let $T_n \to T$ ($T$ is an elliptic operator as well). Let $\lambda$ be the smallest eigenvalue of $T$, and $\varphi$ the positive eigenfunction with norm equal to 1 associated to $\lambda$. Show that $\varphi_n\to\varphi$ as $n\to \infty$.

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In what sense do the operators $T_n$ converge? –  Jose27 Jan 25 '12 at 18:45