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We consider the Laplacian operator $A=\Delta$ with mixed boundary conditions on a bounded domain $\Omega$, and let $D(A)$ be the domain of $A$ corresponding to such boundary conditions.

My questions are :

1) Is $A$ a self-adjoint operator?

2) Does the spectrum of $(A,D(A)) $ decrease to $-\infty$?

3) Does $A$ generate a semigroup of contractions $S(t)$ on $H=L^2(\Omega)?$

4) We define the sets : $ D(A^0)=L^2(\Omega), D(A^1)=D(A), D(A^{k+1}) = \{y\in D(A^k); Ay\in D(A^k)\}.$ Is the injection $D(A^k) \hookrightarrow H^k(\Omega)$ continuous?

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