The Ornstein-Uhlenbeck semigroup in exterior domains
Let \(K\) be a compact in \({\mathbb R}^n\) with \(C^{1,1}\)-boundary, \(\Omega ={\mathbb R}^n\setminus K\) and matrices \(M\in M_n({\mathbb R})\setminus\{0\}\) be given. Then: (i) in \(L^p(\Omega )\), \(1<p<\infty\), operators of the form \(Lu(x):=\Delta u(x)+Mx\cdot \nabla u(x)\), \(x\in\Omega \), with \(D(L)=\{u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)\): \(Mx\cdot \nabla u\in L^p(\Omega )\}\), generate \(C_0\)-semigroups and \(\|e^{tL}\|\leq e^{-(\operatorname{tr} M)t/p}\), \(t\geq0\); (ii) for \(q\in[p,\infty)\), upper bounds of \(\|e^{tL}f\|_q\) and \(\|\nabla e^{tL}f\|_q\) obtained (\(t>0\), \(\|f\|_p=1\)) are similar to those in the case of an ``interior'' bounded domain \(\Omega\). If \(\partial \Omega \) is only Lipschitz, (i) holds for \(p\) close to 2 and, given also the so-called uniform outer ball condition, (i) and (ii) follow for \(1<p\leq q\leq 2\).
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