On adjoint operators associated with boundary value problems

Abstract

For a given linear differential operator, a bilinear identity is established between two arbitrary functions. By using this identity, two boundary value problems, each said to be the adjoint of the other, can be uniquely defined. An unconstrained variational principle follows immediately. The complete system is said to be intrinsic to the given differential operator. The intrinsic identity can be modified by including additional boundary terms of a specific form. Thus, more general boundary value problems and their unique adjoints can be included. The associated variational principles also follow easily. These formulations are concisely summarized by introducing a new adjoint operator. Several physical examples are given that lead to correct adjoint problems and associated variational principles.