Study of a Complete Abstract Differential Equation of Elliptic Type with Variable Operator Coefficients, I

Differential Equation of Elliptic Type with Variable Operator Coefficients, I Angelo FAVINI, Rabah LABBAS, Keddour LEMRABET, and Boubaker-Khaled SADALLAH Universit` degli Studi di Bologna a Laboratoire de Math´ ematiques Appliqu´ ees Dipartimento di Matematica Universit´ du Havre e Piazza di Porta S. U.F.R Sciences et Techniques, B.P 540 Donato 5, 40126 Bologna -- Italy 76058 Le Havre -- France favini@dm.unibo.it rabah.labbas@univ-lehavre.fr Laboratoire EDP et Hist. Maths Lab oraoire AMNEDP Ecole Normale Sup´ erieure Facult´ des Maths, USTHB e 16050 Kouba, Alger -- Algeria BP 32, El Alia, Bab Ezzouar 16111 Alger -- Algeria sadallah@ens-kouba.dz keddourlemrabet@yahoo.fr Received: May 16, 2007 Accepted: August 30, 2007The aim of this first work is the resolution of an abstract complete second order differential equation of elliptic type with variable operator coefficients set in a small length interval. We obtain existence, uniqueness and maximal regularity results under some appropriate differentiability assumptions combining those of Yagi [13] and Da Prato-Grisvard [6]. An example for the Laplacian in a regular domain of R³ will illustrate the theory. A forthcoming work (Part II) will complete the present one by the study of the Steklov-Poincaré operator related to this equation when the length δ of the interval tends to zero