In this paper we give some new results on Sturm-Liouville abstract problems of second-order differential equations of elliptic type in UMD spaces. Existence, uniqueness and maximal regularity of the strict solution are proved using the celebrated Dore-Venni theorem. This work completes the problems studied by Favini, Labbas, Maingot, Tanabe and Yagi under Dirichlet boundary conditions, see .
"Sturm-Liouville problems for an abstract differential equation of elliptic type in UMD spaces." Differential Integral Equations 21 (9-10) 981 - 1000, 2008.