Resolution and optimal regularity for a biharmonic equation withimpedance boundary conditions and some generalizations

Angelo Favini; Rabah Labbas; Keddour Lemrabet; Stéphane Maingot; Hassan D. Sidibé

doi:10.3934/dcds.2013.33.4991

Article Contents

2013, Volume 33, Issue 11&12: 4991-5014. Doi: 10.3934/dcds.2013.33.4991

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Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations

1.
Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna
2.
Laboratoire de Mathématiques Appliquées du Havre, Université du Havre, 25 rue Philippe Lebon, CS 80540, 76058 Le Havre Cedex
3.
Laboratoire AMNEDP, Faculté de Maths USTHB, BP 32, El Alia Bab Ezzouar, 16111 Alger

Received: October 2011

Revised: October 2011

Published: November 2013

Abstract / Introduction Related Papers Cited by

Abstract

In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.

Keywords:

Mathematics Subject Classification: 31A30, 35J40, 31A10, 35Q74, 34K30.

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