Abstract:In this paper we deal with geometrical versions of Hardy type inequalities with additional positive terms in convex domains. The constant $\lambda(\Omega)$ multiplying the additional term depends on the geometry of the multidimensional domain $\Omega$ and the numerical parameters of the problem. The constant (functional) $\lambda(\Omega)$ 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.
From: Ramil Nasibullin [view email]
[v1]
Wed, 17 Jun 2026 18:11:30 UTC (2,734 KB)
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