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

Résumé

The 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.