Calculus of Variations and Geometric Measure Theory
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M. Amar - L. R. Berrone - R. Gianni

Asymptotic expansion for membranes subjected to a lifting force in a part of their boundary

created on 07 Jan 2002
modified on 16 Apr 2004


Published Paper

Inserted: 7 jan 2002
Last Updated: 16 apr 2004

Journal: Asymptotic Analysis
Volume: 36
Pages: 319-343
Year: 2003


The framework of this paper is given by the mixed boundary-value problem \[ \left
\begin{array}{lll} \Delta u(x)=0, & & x\in \Omega u(x)=0, & & x\in \Gamma _{0} {\partial u\over\partial n}(x)=q(x), & & x\in \Gamma _{1} \end{array} \right. \] where $\Omega $ is a plane domain bounded by a regular curve composed by two arcs $\Gamma _{0}$ and $\Gamma _{1}$. Assuming that $\left
\Gamma _{1}\right
=\varepsilon $ and denoting by $u[\varepsilon ]$ the solution to this problem, we study some asymptotic expansions in terms of $\varepsilon $ which are related to $u[\varepsilon ]$. Some connections are presented among these expansions, on one hand, and the geometry of the domain $\Omega $, on the other. In addition, a systematic way is found for computing at the boundary the Ghizzetti's integral that solves the problem.

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