I am looking at the weak Laplacian $\Delta$ on the Sobolevspace $W^{2,2}(0,1)$ and now I am restricting my domain to
- homogeneous Dirichlet boundary conditions (write $\operatorname{D^2_{Dir}}$)
- homogeneous von Neumann boundary conditions (write $\operatorname{D^2_{Neu}}$),
where the boundary conditions are meant to hold for the continuous representative of the function or its derivatie, respectively.
It follows from the Gauss-Green formulas that $\Delta$ restricted to $\operatorname{D^2_{Dir}}$ and also $\Delta$ restricted to $\operatorname{D^2_{Neu}}$ is a symmetric operator (with respect to the $L^2$-inner product). Is $\Delta$ in both cases also self-adjoint?
Yes, both of your operators are selfadjoint. In both cases, there is an orthonormal basis of eigenfunctions. The non-normalized eigenfunctions of the Dirichlet problem are $$ \{ \sin(n\pi x) \}_{n=1}^{\infty}. $$ The non-normalized eigenfunctions of the Neumann problem are $$ \{ \cos(n\pi x) \}_{n=0}^{\infty}. $$ If $\{ s_n \}_{n=1}^{\infty}$ and $\{ c_n \}_{n=0}^{\infty}$ are the normalized eigenfunctions, then \begin{eqnarray*} \Delta_{\mbox{dir}}f &= -\sum_{n=1}^{\infty}n^2\pi^2\langle f,s_n\rangle s_n.\\ \Delta_{\mbox{neu}}f &= -\sum_{n=1}^{\infty}n^2\pi^2\langle f,c_n\rangle c_n. \end{eqnarray*} The domain of $\Delta_{\mbox{dir}}$ consists of all $f$ for which $\{ n^2\pi^2\langle f,s_n\rangle \}_{n=1}^{\infty} \in \ell^2(\mathbb{Z}^+)$. Similarly for the domain of $\Delta_{\mbox{neu}}$.