# A closed extension of the Laplace operator with respect to the supremum norm

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 so-called Wiener-regularity, then it generates an analytic semigroup, see the paper by Arendt and Bénilan.