Study of boundary value and transmission problems in the Hölder spaces

We present new results on the resolution of singular transmission problems in Hölder spaces completing in this way the work in Lp cases given in [A. Favini, R. Labbas, K. Lemrabet, S. Maingot, Study of the limit of transmission problems in a thin layer by the sum theory of linear operators, Rev. Mater. Comput. 18 (1) (2005) 143-176]. Our approach makes use of the impedance notion operator [M.A. Leontovich, Approximate boundary conditions for the electromagnetic field on the surface of a good conductor. Investigations on radiowave propagation, Moscow Acad. Sci. (Part II) (1948)] which leads to obtain direct and simplified problems. We then use the Dunford calculus and some similar techniques as in [R. Labbas, Problèmes aux limites pour une équation différentielle abstraite de type elliptique, Thèse d'état, Université de Nice, 1987; A. El Haial, R. Labbas, On the ellipticity and solvability of abstract second-order differential equation, Electron. J. Differ. Eq. 57 (2001) 1-18; A. Favini, R. Labbas, S. Maingot, H. Tanabe, A. Yagi, Unified study of elliptic problems in Hölder spaces, C. R. Acad. Sci. Paris Ser. 1341 (2005)] in order to prove existence, uniqueness and maximal regularities results.