Advances in Differential Equations

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

Angelo Favini, Davide Guidetti, and Yakov Yakubov

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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.

Article information

Adv. Differential Equations Volume 16, Number 11/12 (2011), 1139-1196.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34G10: Linear equations [See also 47D06, 47D09] 35J25: Boundary value problems for second-order elliptic equations 35J40: Boundary value problems for higher-order elliptic equations 35K20: Initial-boundary value problems for second-order parabolic equations 35K35: Initial-boundary value problems for higher-order parabolic equations


Favini, Angelo; Guidetti, Davide; Yakubov, Yakov. Abstract elliptic and parabolic systems with applications to problems in cylindrical domains. Adv. Differential Equations 16 (2011), no. 11/12, 1139--1196.

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