Let $\Omega \subset \mathbb{R}^N$ be bounded domain with smooth boundary. The first eigenvalue of $-\Delta \colon W_0^{1,2}(\Omega) \cap W^{2,2}(\Omega) \rightarrow L^2(\Omega)$ is strictly positive. This is well known and proved by using the energy method. But I learned that $-\Delta$ can be extended up to $L^1(\Omega)$. In this case, is the first eigenvalue still strictly positive?