Abstract
Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set
for the distance of \(\mathbf{x} \in \Omega \) to the boundary \(\partial \Omega \) of \(\Omega \). Our main objective in this chapter is to gather information about the regularity properties of δ.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Armitage, D.H., Kuran, Ü.: The convexity of a domain and the superharmonicity of the signed distance function. Proc. Am. Math. Soc. 93(4), 598–600 (1985)
Avkhadiev, F.G.: Hardy type inequalities in higher dimensions with explicit estimate of constants. Lobachevskii J. Math. 21, 3–31 (2006). http://ljm.ksu.ru
Balinsky, A., Evans, W.D., Lewis, R.T.: Hardy’s inequality and curvature. J. Funct. Anal. 262, 648–666 (2012) [Available online 13 October 2011]
Barbatis, G., Filippas, S., Tertikas, A.: A unified approach to improved L p Hardy inequalities with best constants. Trans. Am. Math. Soc. 356(6), 2169–2196 (2004)
Brown, R.C., Edmunds, D.D., Rákosník, J.: Remarks on inequalities of Poincaré type. Czech. Math. J. 45, 351–377 (1995)
Bunt, L.H.N.: Bijdrage tot de Theorie der convexe Puntverzamelingen. Thesis, University of Groningen, Amsterdam (1934)
Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton Jacobi equations and optimal control. In: Progress in Nonlinear Differential equations and their Applications, vol. 58, Birkhäuser, Boston (2004)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987) [OX2 GDP]
Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces, and Embeddings. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg/New York (2004)
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC, Boca Raton/London/New York/Washington, DC (1992)
Evans, W.D., Harris, D.J.: Sobolev embeddings for generalized ridged domains, Proc. Lond. Math. Soc. 54, 141–175 (1987)
Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93(3), 418–491 (1959)
Fremlin, D.H.: Skeletons and central sets. Proc. Lond. Math. Soc. 74, 701–720 (1997)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer Classics in Mathematics. Springer, Berlin/Heidelberg/New York (2001). Reprint of the 1998 Edition
Hörmander, L.: Notions of Convexity. Birkhäuser, Boston/Basel/Berlin (1994)
Itoh, J.-I., Tanaka, M.: The Lipschitz continuity of the distance function to the cut locus. Trans. Am. Math. Soc. 353(1), 21–40 (2001)
Kreyszig, E.: Differential Geometry. Dover, New York (1991)
Kuratowski, K.: Topology, vol. II. Academic, New York (1968)
Lewis, R.T., Li, J., Li, Y.: A geometric characterization of a sharp Hardy inequality. J. Funct. Anal. 262(7), 3159–3185 (2012)
Li, Y., Nirenberg, L.: The distance to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Commun. Pure Appl. Math. 18(1), 85–146 (2005)
MacLaurin, C.: A second letter to Martin Folkes, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra. Philos. Trans. 36, 59–96 (1729)
Mantegazza, C., Mennucci, A.C.: Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47, 1–25 (2003)
Motzkin, T.S.: Sur quelques propriétés charactéristiques des ensembles convexes. Atti Real. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Serie VI 21, 562–567 (1935)
Newton, I.: Sive de compositione et resolutione arithmetica liber. Arithmetica universalis (1707)
Psaradakis, G.: L 1 Hardy inequalities with weights. J. Geom. Anal. 23(4), 1703–1728 (2013)
Stein, E.M.: Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton (1970)
Thorpe, J.: Elementary Topics in Differential Geometry. Undergraduate Texts in Mathematics, Springer, New York (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Balinsky, A.A., Evans, W.D., Lewis, R.T. (2015). Boundary Curvatures and the Distance Function. In: The Analysis and Geometry of Hardy's Inequality. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-22870-9_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-22870-9_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22869-3
Online ISBN: 978-3-319-22870-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)