Non-Homogeneous Boundary Value Problems and Applications

1. We describe, at first in a very formaI manner, our essential aim. n Let m be an op en subset of R , with boundary am. In m and on am we introduce, respectively, linear differential operators P and Qj' 0 ~ i ~ 'V. By "e;non-homogeneous boundary value problem"e; we mean a problem of the following type: let f and gj' 0 ~ i ~ 'v, be given in function space s F and G , F being a space"e; on m"e; and the G/ s spaces"e; on am"e; ; j we seek u in a function space u/t "e;on m"e; satisfying (1) Pu = f in m, (2) Qju = gj on am, 0 ~ i ~ 'v])). Qj may be identically zero on part of am, so that the number of boundary conditions may depend on the part of am considered 2. We take as "e;working hypothesis"e; that, for fEF and gjEG , j the problem (1), (2) admits a unique solution u E U/t, which depends 3 continuously on the data . But for alllinear probIems, there is a large number of choiees for the space s u/t and {F; G} (naturally linke d together). j Generally speaking, our aim is to determine families of spaces 'ft and {F; G}, associated in a "e;natural"e; way with problem (1), (2) and con- j venient for applications, and also all possible choiees for u/t and {F; G} j in these families.