Abstract
We consider problems for quite general second-order abstract elliptic and corresponding parabolic equations on the interval $[0,1]$ and the rectangle $[0,T]\times [0,1]$, 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.
Citation
Angelo Favini. Davide Guidetti. Yakov Yakubov. "Abstract elliptic and parabolic systems with applications to problems in cylindrical domains." Adv. Differential Equations 16 (11/12) 1139 - 1196, November/December 2011. https://doi.org/10.57262/ade/1355703114
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