Generation of analytic semigroups by elliptic operators with Dirichlet boundary conditions in a cylindrical domain (Q1434108)

Let \(O=O_1\times (0,l)\) be a cylinder in \({\mathbb R}^{n+2}\), \(O_1\) being an open bounded set in \({\mathbb R}^{n+1}\) with a boundary of class \(C^2\). Let \({\mathcal A}=\sum_{i,j=1}^{n+2}\,a_{i,j}(x)D_{x_i}D_{x_j}+\sum_{i=1}^{n+2}\,a_i(x)D_{x_i}+a_0(x)\) be a linear second-order operator with coefficients \(a_i\), \(a_0\in C({\overline O})\) and \(a_{i,j}\in C^\alpha({\overline O})\) for some \(\alpha \in [0,+\infty)\) and satisfying \(\sum_{i,j=1}^{n+2}\,a_{i,j}(x)\xi_i\xi_j\geq \mu| \xi| ^2\) \(\forall x\in {\overline O}\), \(\forall \xi\in {\mathbb R}^{n+2}\) and some \(\mu \in (0,+\infty)\). Assume that there exists a \(\varphi_0\in (0,\pi)\) such that \[ \begin{multlined} \cos^{-1}\left(- {\sum_{i=1}^{n+1}\, a_{i,n+2}(x^0)\nu_i(x')\over \big[ a_{n+2,n+2}(x^0) \sum_{i,j=1}^{n+1}\,a_{i,j}(x^0) \nu_i(x')\nu_j(x')\big]^{1/2}}\right)\leq \varphi_0,\\ \forall x^0\in \partial O_1\times \{0,l\},\;x^0=(x',x_{n+2}),\end{multlined} \] \(\nu(x')\) denoting the outward normal unit vector at \(x'\in\partial O_1\). Then, if \(\alpha> \max\,\{0,2-\pi/\varphi_0\}\), \({\mathcal A}\), endowed with Dirichlet boundary conditions, generates a continuous semigroup in \(C({\overline O})\). Furthermore, some specific additional regularity results are proved for the solution \(u\) to the spectral problem \[ (\lambda - {\mathcal A})u=f,\quad \text{ in\;} O,\quad u=0\quad \text{ in\;} \partial O,\quad \lambda \in {\mathbb C}, \] in the framework of appropriate \(C^\sigma\)-spaces with negative index. \noindent When \(\varphi\in (\pi/2,\pi)\) and \(\alpha> \max\,\{0,2-\pi/\varphi_0\}\), some regularity results to the previous problem hold also in weighted \(C^\sigma\)-spaces. \noindent Finally, some inclusions are proved for the interpolation spaces between \({\mathcal D}(A)=\{u\in H^2(O)\cap C({\overline O}):{\mathcal A}u\in C({\overline O})\}\) and \(C({\overline O})\).