On an overdetermined weighted differential inequality of Hardy-type of second order

Abstract

The classical one-dimensional integral Hardy's inequality, despite its one-dimensionality, has numerous applications in many branches of mathematics. Beginning in the 1930s, weighted versions of Hardy's inequality began intensively studied, but the first successes, in the sense of fulfillment criteria, were obtained in the years 1969-1970. At present, the one-dimensional weight Hardy inequality, for almost all values of the parameters has been rather well studied. Along with integral inequality, weight differential Hardy inequality occupies an equally important place. Weighted differential Hardy inequality is studied under various boundary conditions at the boundary of a given interval. However, the specified boundary conditions depend on the behavior of the weight functions at the ends of the interval. In addition, the task depends on the finiteness or infinity of the end of the interval, since the integral behaviors of weight functions behave differently. There are various problems here, especially in the redefined case, i.e. when the given boundary conditions are greater than the order of differentiation. In this article, the problem is studied on a finite interval and it is believed that the features of the weight functions are concentrated at one end of the interval and the boundary conditions are overdetermined.