On integral inequalities associated with ordinary regular differential expressions

Abstract

This chapter analyzes integral inequalities associated with ordinary regular differential expressions. The discussion begins with considering a specific integral inequality ∫{ p | f' | 2 + q | f | 2 } ≥ μ ∫ w | f | 2 ( f ∈ D ), where p , q , and w are real-valued coefficients on the closed bounded interval [ a , b ] with p and w non-negative and D is a linear manifold of complex-valued function on [ a , b ] chosen, so that all the three integrals are absolutely convergent. The chapter discusses the so-called regular case of this inequality—that is, when the coefficients 1/ p , q , and w are all integrable on [ a , b ]. The use of the term regular in this case is in accordance with a similar usage of this word in the theory of ordinary differential operators that plays a fundamental role in determining the parameters of the inequality (1.1). The difficulty in proving equation lies in the fact that whilst the inequality is required on the maximal set D , the parameters of the inequality, that is, the best-possible value of μ and the resulting cases of equality, are determined by a self-adjoint differential operator T with domain D ( T ).