Lp-theory for second-order elliptic operators with unbounded coefficients

Second-order elliptic operators with unbounded coefficients of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Au := -{\rm div}(a\nabla u) + F . \nabla u + Vu}$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p}(\mathbb{R}^{N}) (N \in \mathbb{N}, 1 < p < \infty)}$$\end{document} are considered, which are the same as in recent papers Metafune et al. (Z Anal Anwendungen 24:497–521, 2005), Arendt et al. (J Operator Theory 55:185–211, 2006; J Math Anal Appl 338: 505–517, 2008) and Metafune et al. (Forum Math 22:583–601, 2010). A new criterion for the m-accretivity and m-sectoriality of A in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{p}(\mathbb{R}^{N})}$$\end{document} is presented via a certain identity that behaves like a sesquilinear form over Lp × Lp'. It partially improves the results in (Metafune et al. in Z Anal Anwendungen 24:497–521, 2005) and (Metafune et al. in Forum Math 22:583–601, 2010) with a different approach. The result naturally extends Kato’s criterion in (Kato in Math Stud 55:253–266, 1981) for the nonnegative selfadjointness to the case of p ≠ 2. The simplicity is illustrated with the typical example \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Au = -u\hspace{1pt}'' + x^{3}u\hspace{1pt}' + c |x|^{\gamma}u}$$\end{document} in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^p(\mathbb{R})}$$\end{document} which is dealt with in (Arendt et al. in J Operator Theory 55:185–211, 2006; Arendt et al. in J Math Anal Appl 338: 505–517, 2008).