Advances in Differential Equations

Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, and Hassan Diaramouna Sidibé

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In this work, we present a new result ofexistence, uniqueness and maximal regularity for the restriction solutions of the bilaplacian transmission problem set in the juxtaposition of two rectangular bodies. The study is performed in the space $L^{p}( (-1, 0) \cup(0, \delta) ;X),$ $1 < p < \infty,$ where $\delta $ is a small parameter which is destined to tend to zero and $X$ is a UMD Banach space. The geometry of the bodies allows us to find an explicit representation of the solutions in virtue of the operational Dunford calculus. We then use essentially the famous Dore-Venni theorem among others for the analysis of the solutions.

Article information

Adv. Differential Equations Volume 15, Number 1/2 (2010), 43-72.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35C15: Integral representations of solutions 35J30: Higher-order elliptic equations [See also 31A30, 31B30] 35J40: Boundary value problems for higher-order elliptic equations 45N05: Abstract integral equations, integral equations in abstract spaces


Favini, Angelo; Labbas, Rabah; Lemrabet, Keddour; Maingot, Stéphane; Sidibé, Hassan Diaramouna. Transmission problem for an abstract fourth-order differential equation of elliptic type in UMD spaces. Adv. Differential Equations 15 (2010), no. 1/2, 43--72.

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