On Sufficient Conditions for the Weak Extremum of a Quadratic Functional with the Stieltjes Integral

Abstract

The paper studies a quadratic functional containing second derivatives, which is equal to the potential energy of a system consisting of a chain of pivotally connected rods, and placed in an inhomogeneous medium with localized features. The resulting boundary value problem, as a necessary condition for the extremum, has non-smooth solutions. We overcome the difficulties associated with the loss of smoothness in solutions of emerging boundary value problems using the concept of a point-by-point interpretation of a differential equation, which has shown its effectiveness in analyzing solutions of differential equations with non-smooth solutions not only of the second order, but also of a higher one. The latter circumstance has a simple explanation: at each point, the equations are regarded as a relationship between the value of the solution and its derivatives up to a certain order, in contrast to the theory of generalized functions, where it is understood as the equality of two functionals over the basic space. Sufficient conditions for the weak extremum of a quadratic functional with Radon-Nicodemus derivatives are obtained.