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Boundary Curvatures and the Distance Function

  • Alexander A. Balinsky
  • W. Desmond Evans
  • Roger T. Lewis
Part of the Universitext book series (UTX)

Abstract

Let \(\Omega \) be an open subset of \(\mathbb{R}^{n},\ n \geq 2,\) with non-empty boundary, and set
$$\displaystyle{\delta (\mathbf{x}):=\inf \{ \vert \mathbf{x} -\mathbf{y}\vert: \mathbf{y} \in \mathbb{R}^{n}\setminus \Omega \}}$$
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.

Keywords

Distance Function Boundary Curvature Principal Curvature Principal Direction Radon Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 8.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 10.
    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
  3. 20.
    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]Google Scholar
  4. 23.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 32.
    Brown, R.C., Edmunds, D.D., Rákosník, J.: Remarks on inequalities of Poincaré type. Czech. Math. J. 45, 351–377 (1995)zbMATHGoogle Scholar
  6. 33.
    Bunt, L.H.N.: Bijdrage tot de Theorie der convexe Puntverzamelingen. Thesis, University of Groningen, Amsterdam (1934)Google Scholar
  7. 35.
    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)Google Scholar
  8. 48.
    Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Oxford University Press, Oxford (1987) [OX2 GDP]zbMATHGoogle Scholar
  9. 49.
    Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces, and Embeddings. Springer Monographs in Mathematics. Springer, Berlin/Heidelberg/New York (2004)zbMATHCrossRefGoogle Scholar
  10. 52.
    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)zbMATHGoogle Scholar
  11. 53.
    Evans, W.D., Harris, D.J.: Sobolev embeddings for generalized ridged domains, Proc. Lond. Math. Soc. 54, 141–175 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 58.
    Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93(3), 418–491 (1959)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 65.
    Fremlin, D.H.: Skeletons and central sets. Proc. Lond. Math. Soc. 74, 701–720 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 68.
    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 EditionGoogle Scholar
  15. 80.
    Hörmander, L.: Notions of Convexity. Birkhäuser, Boston/Basel/Berlin (1994)zbMATHGoogle Scholar
  16. 81.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 90.
    Kreyszig, E.: Differential Geometry. Dover, New York (1991)Google Scholar
  18. 93.
    Kuratowski, K.: Topology, vol. II. Academic, New York (1968)Google Scholar
  19. 107.
    Lewis, R.T., Li, J., Li, Y.: A geometric characterization of a sharp Hardy inequality. J. Funct. Anal. 262(7), 3159–3185 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 108.
    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)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 114.
    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)Google Scholar
  22. 115.
    Mantegazza, C., Mennucci, A.C.: Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47, 1–25 (2003)MathSciNetCrossRefGoogle Scholar
  23. 120.
    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)Google Scholar
  24. 123.
    Newton, I.: Sive de compositione et resolutione arithmetica liber. Arithmetica universalis (1707)Google Scholar
  25. 127.
    Psaradakis, G.: L 1 Hardy inequalities with weights. J. Geom. Anal. 23(4), 1703–1728 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  26. 139.
    Stein, E.M.: Singular Integrals and Differentiability properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  27. 143.
    Thorpe, J.: Elementary Topics in Differential Geometry. Undergraduate Texts in Mathematics, Springer, New York (1994)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Alexander A. Balinsky
    • 1
  • W. Desmond Evans
    • 1
  • Roger T. Lewis
    • 2
  1. 1.School of MathematicsCardiff UniversityCardiffUK
  2. 2.Department of MathematicsUniversity of Alabama at BirminghamBirminghamUSA

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