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I am looking at the weak Laplacian $\Delta$ on the Sobolevspace $W^{2,2}(0,1)$ and now I am restricting my domain to

  1. homogeneous Dirichlet boundary conditions (write $\operatorname{D^2_{Dir}}$)
  2. 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?

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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 $\Delta_{\mbox{neu}}$.

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    $\begingroup$ Sorry, I do have one further questions concerning this issue. I hope there is just one point which I do not see, but for the self adjointness of that is left to show is that $\operatorname{dom}(\Delta_{dir}^{*}) \subseteq D^{2}_{Dir}$ (and the same for Neumann). How can I see this directly from your spectral decomposition? $\endgroup$ Mar 16, 2018 at 17:12
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    $\begingroup$ For any sequence of real numbers $\lambda_n$ and any orthonormal set $\{ e_n \}$, the operator $Lf = \sum_{n=1}^{\infty}\lambda_n \langle f,e_n\rangle e_n$ is selfadjoint on its natural domain consisting of all $f$ for which $\sum_{n=1}^{\infty}\lambda_n^2|\langle f,e_n\rangle|^2 < \infty$. $\endgroup$ Mar 16, 2018 at 18:07

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