On solution of partial differential equations by the Hahn-Banach theorem

Abstract

Abstract : A means of construction is presented for the Green's and Neuman n's functions of a linear ellipitc partial differential equation by a method based on linear functionals. The construction depends essentially on the Hahn-Banach extension theorem (Theorle des Operations Lineaires, Warsaw, 1932) which was used in a similar connection by Lax (Proc. Amer. Math. Soc. 3:526-531, 1952). Lax's approach differs in that, while his proof centers about a bounded linear functional based on inhomogeneous boundary conditions, this proof centers about a functional based on the solution of an inhomogeneous differential equation. This latter point of view has the advantage that in constructing the Green's and Neumann's functions only a fundamental solution of the differential equation is needed in specific instances, and success is possible with merely a parametrix. These considerations are also advantageous for domains with general boundaries and for Riemannian manifolds. The proof of the Hahn-Banach theorem in the form used does not require transfinite induction, since the Banach space of continous functions is separable.