Abstract
We study to what extent rearrangements preserve the integrability properties of higher order derivatives. It is well known that the second order derivatives of the rearrangement of a smooth function are not necessarily inL 1. We obtain a substitute for this fact. This is done by showing that the total curvature for the graph of the rearrangement of a function is bounded by the total curvature for the graph of the function itself.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cianchi, A., Second-order derivatives and rearrangements,Duke Math. J. 105 (2000), 355–385.
Hardy, G. H., Littlewood, J. E. andPólya, G.,Inequalities, Cambridge Univ. Press, Cambridge, 1952.
Milnor, J. W., On the total curvature of knots,Ann. of Math. 52 (1950), 248–257.
Pólya, G. andSzegő, G.,Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 1951.
Talenti, G., Assembling a rearrangement,Arch. Rational Mech. Anal. 98 (1987), 285–293.
Author information
Authors and Affiliations
Additional information
This posthumous paper was prepared for publication by Vilhelm Adolfsson and Peter Kumlin.
The author was supported by a grant from the Swedish Natural Science Research Council.
Rights and permissions
About this article
Cite this article
Dahlberg, B.E.J. Total curvature and rearrangements. Ark. Mat. 43, 323–345 (2005). https://doi.org/10.1007/BF02384783
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02384783