Let $\Omega\subseteq\mathbb R^n$ be a bounded domain, $H:=W_0^{1,2}(\Omega)$ be the Sobolev space, $C^{2,\alpha}(\Omega)$ be the Hölder space for some $\alpha\in (0,1]$ and $\langle\;\cdot\;,\;\cdot\;\rangle_2$ be the $L^2(\Omega)$ scalar product. Let's consider the eigenvalue problem for $-\Delta$ in $\Omega$, i.e. $$-\Delta u=\lambda u\;.\tag{1}$$ Let $\Theta\in H$ and $\lambda_1>0$ be the first weak eigenfunction and eigenvalue, respectively, i.e. $(\Theta,\lambda_1)$ being a solution of $$\langle\nabla u,\nabla\varphi\rangle_2=\lambda\langle u,\varphi\rangle_2\;\;\;\text{for all }u\in H\tag{2}$$ such that there is no other solution $(u,\lambda)$ of $(2)$ with $\lambda<\lambda_1$.
Now, I've read that if the boundary $\partial\Omega$ is a $C^{2,\alpha}(\Omega)$-submanifold, then
- $\Theta\in C^{2,\alpha}(\Omega)$
- $\Theta>0$
- $\Theta$ is a (strong) eigenfunction with eigenvalue $\lambda_1$, i.e. $(\Theta,\lambda_1)$ is a solution of $(1)$
How do I need to understand this? $\Theta$ is an equivalence class of functions. So, what's is meant (especially) by $(1.)$ and $(2.)$?