The Ornstein-Uhlenbeck semigroup in exterior domains

摘要

Let Ω be an exterior domain in $mathbb{R}^n .$ It is shown that Ornstein-Uhlenbeck operators L generate C 0-semigroups on L p (Ω) for p ∈ (1, ∞) provided ∂Ω is smooth. The method presented also allows to determine the domain D(L) of L and to prove L p − L q smoothing properties of e tL . If ∂Ω is only Lipschitz, results of this type are shown to be true for p close to 2.