On Poisson operators and Dirichlet-Neumann maps in \(H^s\) for divergence form elliptic operators with Lipschitz coefficients (Q2790728)

This paper deals with second-order uniformly elliptic operator \(A\) of divergence form in \(\mathbb{R}^{d+1}\), whose coefficients \(a_{i,j}\) depend on \(x\in\mathbb{R}^d\). At first the authors define the Poisson semigroup associated with \(A\) as the solution to the Dirichlet problem in \(\mathbb{R}^{d+1}_+\). Then the Dirichlet-Neumann map associated with \(A\) is defined through an appropriate sesquilinear form.NEWLINENEWLINE It is shown in Theorem 1.2 that the Poisson semigroup in \(H^{1/2}(\mathbb{R}^d)\) is extended as a strongly continuous and analytic semigroup in \(L^2(\mathbb{R}^d)\). Its generator \(-{\mathcal P}_A\) called the Poisson operator satisfies \({\mathcal D}_{L^2}({\mathcal P}_A)= H^1(\mathbb{R}^d)\) and \(I+{\mathcal P}_A\) admits a bounded \(H^\infty(\Sigma_\varphi)\) calculus in \(L^2(\mathbb{R}^d)\) for some \(\varphi\in (0,{\pi\over 2})\). Here \(\Sigma_\varphi\) is an open sector in \(\mathbb{C}^1\) with angle \(\varphi\). The case when \(s\in[0,1]\) and \(H^s(\mathbb{R}^d)\) stands instead of \(L^2(\mathbb{R}^d)\) is considered, too.NEWLINENEWLINE In Theorem 1.4 a factorization formula for \(A\) in terms of \({\mathcal P}_A\) is given.