Abstract elliptic and parabolic systems with applications to problems in cylindrical domains

We consider problems for quite general second-order abstract elliptic and corresponding parabolic equations on the interval [0; 1] and the rectangle [0,T]×[0,1], respectively. R-boundedness estimates of solutions of abstract boundary-value problems for elliptic equations with a parameter are established, in contrast to standard norm-bounded estimates. The results are applied to obtain L^p-maximal regularity for corresponding parabolic systems. In applications, the coefficient A(x) of the solution u can be 2m-order elliptic operators with suitable boundary conditions, while the coefficient B(x) of the first-order derivative of the solution D_xu can be interpreted as an m-order differential operator. The corresponding applications to PDEs are presented.