We consider bounded Lipschitz domains Ω in ℝ n . It is shown that the Dirichlet-Laplacian generates an analytic C 0-semigroup on L p (Ω) for p in an interval around 2 and that the corresponding Cauchy problem has the maximal L q -regularity property. We then prove that for bounded or exterior Lipschitz domains Ornstein-Uhlenbeck operators A generate C 0-semigroups in the same p-interval. The method, also allows to determine the domain D(A) of A and, if Ω satisfies an outer ball condition, allows to show L p -L q -smoothing properties of the semigroups
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