Generalized solutions of the boundary value problems

Abstract

When we seek to extend the field of validity of the existence theorems proved in the preceding chapter by reducing the hypotheses there admitted for the data of the problem, we are conducted in quite a natural way to consider as solutions of a boundary value problem functions which satisfy only in part the properties specified in § 4, these properties being replaced by others which are less restrictive. The generalized solutions of the various boundary value problems, which we thus come to consider, can be defined in different ways, which differ from one another according as the major scope of the hypotheses which we want to follow relate to the coefficients of the equation, or else to the domain, or even to the boundary conditions. In this chapter we shall occupy ourselves with those existence theorems for generalized solutions which can be proved by having recourse to the theory of integral equations or by putting to service other procedures based ultimately on the Hahn-Banach-Ascoli theorem. We shall also take into consideration Perron’s method of super- and subfunctions and we shall give some idea of the methods of the minimum. Other types of generalized solutions, the study of which requires different procedures, will be considered instead in Chapter V (§§ 37, 38, 39).