Notes on Generalized Boundary Value Problems in Banach Spaces. I. Adjoint and Extension Theory.

Multipoint boundary conditions arise in the theory of beams or plates with interior point loads, and also the mathematical theory of splines. Interface conditions arise in problems of diffusion through parallel slabs with different properties e.g., nuclear reactors or the study of shock waves. Adjoints of such differential operators also are encountered when one attempts to derive Euler-Lagrange equations for constrained minimization problems. Our method is very general and is designed to work for partial differential, integral and functional differential operators as well as differential operators. Part I presents some of the abstract machinery to solve the problem. Part II will apply this machinery to concrete and applied problems of the type mentioned above.