In the following let $\Omega$ be an open and bounded domain in $\mathbb{R}^d$ with smooth boundary. As far as I understood, for the Laplacian $\Delta: H_0^1(\Omega)\to L^2(\Omega)$ with Dirichlet boundary conditions there exists a eigenpair $(e_n)$ and $(\lambda_n)$, such that $(e_n)$ forms a ONS of $L^2(\Omega)$.
Question: Is it right, that $e_n$ then also has zero boundary values, $e_n(x)=0$ for $x\in\partial\Omega$?
Now, can I consider the fractionally defined Sobolev space $\tilde{H}^s(\Omega)$ consisting of all functions $f\in L^2(\Omega)$, such that $\sum_{n\in\mathbb{N}}\lambda_n^{2s}\langle f,e_n\rangle_{L^2}^2 < \infty$
Question: Do functions in $\tilde{H}^s(\Omega)$ also have zero boundary values ? If yes, does it hold for all $s\geq 0$? In other words, do we have the embedding $\tilde{H}^s(\Omega)\hookrightarrow H_0^s(\Omega)$, where $H_0^s(\Omega)$ is the 'usual' Sobolev space of all weak differentiable functions up to order $s$ (defined by interpolation for non-integer $s$) with zero boundary values.
Question For $s\geq 0$ is $\tilde{H}^{-s}(\Omega)$ well-defined ? Is it the dual space of $\tilde{H}^s(\Omega)$ ?
thanks in advance!!