Abstract
We present the area and coarea formulas for Lipschitz maps, valid for general volume densities. We apply these formulas to derive an anisotropic tube formula for hypersurfaces in \({{\mathbb{R}}^n}\) and to give a short euclidean proof of the anisotropic Sobolev inequality. A discussion about the first variation of the anisotropic area is also included.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvino A., Ferone V., Trombetti G., Lions P-L.: Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 14(2), 275–293 (1997)
Ambrosio L., Kirchheim B.: Rectifiable sets in metric and Banach spaces. Math. Ann. 318, 527–555 (2000)
Álvarez Paiva J.C., Thompson A.C.: Volumes on Normed and Finsler Spaces. A sampler of Riemann-Finsler Geometry, MSRI Pub., vol. 50. pp. 1–48. Cambridge Univ. Press, Cambridge (2004)
Andrews B.: Volume-preserving anisotropic mean curvature flows. Indiana Univ. Math. J. 50(2), 783–827 (2001)
Chambolle, A., Lisini, S., Lussardi, L.: A remark on the anisotropic outer Minkowski content. http://arxiv.org/abs/1203.5190
Cordero-Erausquin D., Nazaret B., Villani C.: A mass transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)
Evans L., Gariepy R.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)
Federer H.: Geometric Measure Theory. Grundlehren der Mathematischen Wissenschaften. Springer, New York (1969)
Federer H.: Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)
Federer H., Fleming W.: Normal and integral currents. Ann. Math. 72(3), 458–520 (1960)
Figalli, A., Maggi, F., Pratelli, A.: A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182(1), 167–211 (2010)
Greub W., Halperin S., Vanstone R.: Connections, Curvature and Cohomology, vol. 1. Academic Press, San Diego (1972)
Kirchheim B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. AMS 121(1), 113–123 (1994)
Koiso M., Palmer B.: Geometry and stability of surfaces with constant anisotropic mean curvature. Indiana Univ. Math. J. 54(6), 1817–1852 (2005)
Milman V., Schechtman G.: Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Mathematics 1200. Springer, Berlin (1986)
Palmer B.: Stability of the Wulff shape. Proc. AMS 126(12), 3661–3667 (1998)
Schneider R.: Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia of Mathematics and its Applications 44. Cambridge University Press, Cambridge (1993)
Shen Z.: Lectures on Finsler Geometry. World Scientific Publishing Co., Singapore (2001)
Simon, L.: Lectures on Geometric Measure Theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, 1983
Taylor J.: Crystalline variational problems. Bull. AMS. 84, 568–588 (1978)
van Schaftingen J.: Anisotropic symmetrization. Ann. Inst. H. Poincar Anal. Non Linaire 23, 539–565 (2006)
Zhang G.: The affine Sobolev inequality. J. Differ. Geom. 53(1), 183–202 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cibotaru, D., de Lira, J. A note on the area and coarea formulas for general volume densities and some applications. manuscripta math. 149, 471–506 (2016). https://doi.org/10.1007/s00229-015-0779-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-015-0779-x