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 $C_0(X)$?
The paper shows this even for unbounded domains, along with the fact that if the regularity assumption is not satisfied, then the Dirichlet-Laplace operator has empty resolvent (hence it is not even a semigroup generator).
For bounded domains and well-posedness (=semigroup generation), see Gilbarg-Trudinger, Section 2.8.