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Selected results and open problems on Hardy–Rellich and Poincaré–Friedrichs inequalities

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Abstract

In this paper we give a survey of selected results and open problems on integral inequalities of Mathematical Physics connected with the papers of V. Maz’ya, S. Fillippas, A. Tertikas, R. Osserman, A. Ancona, H. Brezis, M. Marcus, Y. Pinchover, E. B. Davies, A. Laptev, J. L. Fernández, J. M. Rodríguez, P. Caldiroli, R. Musina, A. A. Balinsky, W. D. Evans, R. T.  Lewis, R. G. Nasibullin, I. K. Shafigullin, the author and other mathematicians. In addition, we give some new examples and present non-linear relationships between global numerical characteristics of domains in the Euclidean space of dimension \(n\ge 2\).

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Acknowledgements

The author is grateful to the referee for his comments, corrections and suggestions.

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  1. Lobachevskii Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya st. 35, Kazan, Tatarstan, 420008, Russia

    Farit Avkhadiev

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This research is supported by the Russian Science Foundation under Grant no. 18-11-00115.

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Avkhadiev, F. Selected results and open problems on Hardy–Rellich and Poincaré–Friedrichs inequalities. Anal.Math.Phys. 11, 134 (2021). https://doi.org/10.1007/s13324-021-00568-3

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