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 $C_0(X)$?

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. 

