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

Mathematics > Functional Analysis [Submitted on 17 Jun 2026] Title:A proof of the Avkhadiev-Wirths conjecture on Brezis-Marcus constants View PDF HTML (experimental)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. Submission history From: Ramil Nasibullin [view email][v1] Wed, 17 Jun 2026 18:11:30 UTC (2,734 KB) References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.