A proof of the Avkhadiev-Wirths conjecture on Brezis-Marcus constants

Abstract

In this paper we deal with geometrical versions of Hardy type inequalities with additional positive terms in convex domains. The constant $λ(Ω)$ multiplying the additional term depends on the geometry of the multidimensional domain $Ω$ and the numerical parameters of the problem. The constant (functional) $λ(Ω)$ is called Brezis-Marcus constant. In 2010, F.G. Avkhadiev and K.-J. Wirths proposed the hypothesis that among all $n$-dimensional domains with given inradius the maximum of the best Brezis-Marcus constant is achieved for the $n$-dimensional ball of radius. Using one dimensional Hardy type inequalities we proved the Avkhadiev-Wirths conjecture on Brezis-Marcus constants in the cases $n=2$ and $n\geq 4$. The sharp constants are solutions of the equation in terms of special functions and fixed eigenvalues of the Sturm-Liouville differential operators. The corresponding eigenfunctions in the $2$-d case are spheroidal wave functions and for dimensions greater than or equal to $4$ are confluent Heun functions. New properties of the Heun functions are established and their zeros are found. We provide Python code for calculating sharp constants.