Resumen de Study of the limit of transmission problems in a thin layer by the sum theory of linear operators

Angelo Favini, Keddour Lemrabet, Labbas Rabah, Stéphane Maingot

• We consider a family (Pd), where d is a small positive parameter, of singular elliptic transmission problems in the juxtaposition O =]-1, d [x G of two bodies, the cylindric medium O=] - 1,O[xG and the thin layer O=]O, d [xG. It in assumed that the coefficient in O in 1/ d. Such problems model for instance heat propagation between the body O -, the layer O + (when supposed with infinite conductivity), and the ambient space. After performing a recalling in the thin layer to transform the problem in the fixed domain ]- 1, 1 [x 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 L0 spaces, p> 1. We deduce that the family of solution u~ converges in Lt to a function u in the case of second member in Lp and convergent in Wn+20P for a second member in W1+20p, (0 e]0, 1/2]). We then prove that the restriction of the limit u to ]- 1,0[xG is in fact the solution to an elliptic problem on ]- 1, O]xG with a boundary condition of Ventcel's type and it has an optimal regularity.