Laplacian dissipative?

Question

is it true that the Laplacian $\Delta:=\frac{d^2}{dx^2}$ on $(0,1)$ with Neumann boundary conditions is dissipative on $C[0,1]?$

For this we have to show that there is for any $x \in D(\Delta)$a $x' \in C[0,1]'$ such that $x'(x)=\left\lVert x\right\rVert^2=\left\lVert x'\right\rVert^2$ and $$\Re \langle \Delta x,x' \rangle \le 0.$$

Does anybody know how to choose this x'?

Answer 1

See the example of Section II.3.30 in

Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2000). ZBL0952.47036.