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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 &#948; of the interval tends to zero

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