Let $X$ be a bounded connected open subset of the $n$dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support in $X$. What is the closure of this operator in the space $C_0(X)$ endowed with the supremum norm? Does its closure generate a strongly continuous semigroup on $C_0(X)$?

1$\begingroup$ It is good to mention that you have already asked this at MSE, did not get reaction, and reposted here, see math.stackexchange.com/questions/438166/… $\endgroup$ – András Bátkai Jul 9 '13 at 20:16
If $X$ has the socalled Wienerregularity, then it generates an analytic semigroup, see the paper by Arendt and Bénilan.
The paper shows this even for unbounded domains, along with the fact that if the regularity assumption is not satisfied, then the DirichletLaplace operator has empty resolvent (hence it is not even a semigroup generator).
For bounded domains and wellposedness (=semigroup generation), see GilbargTrudinger, Section 2.8.

$\begingroup$ Thank you. But still I am not sure that the space of infinitely differentiable functions with compact support in X is a core for the Dirichlet Laplacian (If we suppose that the domain X is regular). If yes, this space should be dense in the domain of the laplacian with respect to the graph norm. $\endgroup$ – user36746 Jul 9 '13 at 13:38

$\begingroup$ @user36746: It seems to be, but I am not sure at the moment. I am sure that it is the case for sufficiently nice domains, but have to think about the general case. $\endgroup$ – András Bátkai Jul 9 '13 at 20:28