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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 "non-homogeneous boundary value problem" 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" on m" and the G/ s spaces" on am" ; j we seek u in a function space u/t "on m" 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 "working hypothesis" 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 "natural" 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.
Contents
0.- 13. Intersection Interpolation.- 14. Holomorphic Interpolation.- 15. Another Intrinsic Definition of the Spaces [X, Y]0.- 16. Compactness Properties.- 17. Comments.- 18. Problems.- 2 Elliptic Operators. Hilbert Theory.- 1. Elliptic Operators and Regular Boundary Value Problems.- 2. Green's Formula and Adjoint Boundary Value Problems.- 3. The Regularity of Solutions of Elliptic Equations in the Interior of ?.- 4. A priori Estimates in the Half-Space.- 5. A priori Estimates in the Open Set ? and the Existence of Solutions in Hs(?)-Spaces, with Real s ? 2m.- 6. Application of Transposition: Existence of Solutions in Hs(?)-Spaces, with Real s ? 0.- 7. Application of Interpolation: Existence of Solutions in Hs(?)-Spaces, with Real s, 0 < s < 2m.- 8. Complements and Generalizations.- 9. Variational Theory of Boundary Value Problems.- 10. Comments.- 11. Problems.- 3 Variational Evolution Equations.- 1. An Isomorphism Theorem.- 2. Transposition.- 3. Interpolation.- 4. Example: Abstract Parabolic Equations, Initial Condition Problem (I).- 5. Example: Abstract Parabolic Equations, Initial Condition Problem (II).- 6. Example: Abstract Parabolic Equations, Periodic Solutions.- 7. Elliptic Regularization.- 8. Equations of the Second Order in t.- 9. Equations of the SecondOrder in t; Transposition.- 10. Schroedinger Type Equations.- 11. Schroedinger Type Equations; Transposition.- 12. Comments.- 13. Problems.
