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

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≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}.