RMC - Vol. 18 Num. 1, Favini et al.

Standard reference:
$% A. Favini, R. Labbas, K. Lemrabet, and S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mat. Complut. 18 (2005), no. 1, 143–176.%$

Study of the Limit of Transmission Problems in a Thin Layer by the Sum Theory of Linear Operators

Angelo FAVINI, Rabah LABBAS,
Keddour LEMRABET, and Stéphane MAINGOT

Università degli Studi di Bologna Dipartimento di Matematica Piazza di Porta S. Donato, 5 40126 Bologna — Italy favini@dm.unibo.it	Laboratoire de Mathématiques Appliquées Université du Havre U.F.R Sciences et Techniques, B.P 540 76058 Le Havre — France rabah.labbas@univ-lehavre.fr stephane.maingot@univ-lehavre.fr

Département de Maths.
USTHB, BP 32, El Alia
Bab Ezzouar, 16111 Alger — Algeria

keddourlemrabet@wissal.dz

Received: April 19, 2004
Accepted: June 27, 2004

ABSTRACT

We consider a family $(Pd)$ , where $d$ is a small positive parameter, of singular elliptic transmission problems in the juxtaposition $_O_d =]- 1,d[× G$ of two bodies, the cylindric medium $_O_- =]- 1,0[×G$ and the thin layer $_O_d+ =]0,d[×G$ . It is assumed that the coefficient in $_O_d+$ is $1/d$ . Such problems model for instance heat propagation between the body $_O_-$ , the layer $_O_d+$ (when supposed with infinite conductivity), and the ambient space. After performing a rescaling in the thin layer to transform the problem in the fixed domain $]- 1,1[× G$ , it is shown that the sum of operators’ method by Da Prato and Grisvard works and gives an existence and uniqueness result in the framework $p L$ spaces, $p> 1$ . We deduce that the family of solutions $d u$ converges in $p L$ to a function $u$ in the case of second member in $p L$ and converges in $1+2h,p W$ for a second member in $2h,p W$ , ( $h (- ]0,1/2[$ ). We then prove that the restriction of the limit $u$ to $]- 1,0[×G$ is in fact the solution to an elliptic problem on $]- 1,0[×G$ with a boundary condition of Ventcel’s type and it has an optimal regularity.

Key words: sums of linear operators, elliptic problems, interpolation spaces, Ventcel’s problem.
2000 Mathematics Subject Classification: 34K06, 34K10, 34K30.