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Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 591–621 | Cite as

Pointwise differentiability of higher order for sets

  • Ulrich MenneEmail author
Article
  • 45 Downloads

Abstract

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher-order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.

Keywords

Higher-order pointwise differentiability Rectifiability Rademacher–Stepanov type theorem Stationary integral varifold 

Mathematics Subject Classification

Primary: 51M05 Secondary: 26B05 49Q20 

Notes

Acknowledgements

The author would like to thank Mario Santilli for reading part of the manuscript and for bringing a series of papers of Isakov to his attention, Dr. Sławomir Kolasiński for helping him to become acquainted with some of these results available only in Russian, and Dr. Yangqin Fang for pointing him to [11]. The initial version of this paper (see https://arxiv.org/abs/1603.08587v1) was written while the author worked at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and the University of Potsdam. The subsequent revision was made while the author worked at the University of Leipzig and the Max Planck Institute for Mathematics in the Sciences.

References

  1. 1.
    Allard, W.K., Almgren Jr., F.J.: The structure of stationary one dimensional varifolds with positive density. Invent. Math. 34(2), 83–97 (1976).  https://doi.org/10.1007/BF01425476 MathSciNetzbMATHGoogle Scholar
  2. 2.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Modern Birkhäuser Classics. Birkhäuser Boston Inc., Boston (2009).  https://doi.org/10.1007/978-0-8176-4848-0. Reprint of the 1990 editionGoogle Scholar
  3. 3.
    Allard, W.K.: On the first variation of a varifold. Ann. of Math. 2(95), 417–491 (1972).  https://doi.org/10.2307/1970868 MathSciNetzbMATHGoogle Scholar
  4. 4.
    Almgren, Jr. F.J.: Almgren’s Big Regularity Paper, Volume 1 of World Scientific Monograph Series in Mathematics. World Scientific Publishing Co. Inc., River Edge (2000). \(Q\)-valued functions minimizing Dirichlet’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2, With a preface by Jean E. Taylor and Vladimir Scheffer.  https://doi.org/10.1142/9789812813299
  5. 5.
    Anzellotti, G., Serapioni, R.: \({\cal{C}}^k\)-rectifiable sets. J. Reine Angew. Math. 453, 1–20 (1994).  https://doi.org/10.1515/crll.1994.453.1 MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beer, G.: On convergence of closed sets in a metric space and distance functions. Bull. Aust. Math. Soc. 31(3), 421–432 (1985).  https://doi.org/10.1017/S0004972700009370 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brakke, K.A.: The Motion of a Surface by Its Mean Curvature, Volume 20 of Mathematical Notes. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  8. 8.
    Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa (3) 18, 137–160 (1964)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L., Crandall, M.G., Kocan, M., Święch, A.: On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm. Pure Appl. Math. 49(4), 365–397 (1996). http://doi.org/10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V
  10. 10.
    Calderón, A.-P., Zygmund, A.: Local properties of solutions of elliptic partial differential equations. Studia Math. 20, 171–225 (1961)MathSciNetzbMATHGoogle Scholar
  11. 11.
    David, G.: Limits of Almgren quasiminimal sets. In: Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001), volume 320 of Contemporary Mathematics, pp. 119–145. American Mathematical Society, Providence, (2003).  https://doi.org/10.1090/conm/320/05603
  12. 12.
    Dunford, N., Schwartz, J.T.: Linear Operators. I. General Theory. With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7. Interscience Publishers, Inc., New York, London (1958)Google Scholar
  13. 13.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, revised edition. CRC Press, Boca Raton (2015)Google Scholar
  14. 14.
    Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969).  https://doi.org/10.1007/978-3-642-62010-2 Google Scholar
  15. 15.
    Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1975). A modern treatment of the theory of functions of a real variable, Third printing, Graduate Texts in Mathematics, No. 25.  https://doi.org/10.1007/978-3-642-88044-5
  16. 16.
    Ilmanen, T.: A strong maximum principle for singular minimal hypersurfaces. Calc. Var. Partial Differential Equations 4(5), 443–467 (1996).  https://doi.org/10.1007/s005260050049 MathSciNetzbMATHGoogle Scholar
  17. 17.
    Isakov, N.M.: On a global property of approximately differentiable functions. Mathematical Notes 41(4), 280–285 (1987).  https://doi.org/10.1007/BF01137673 MathSciNetzbMATHGoogle Scholar
  18. 18.
    Isakov, N.M.: On a global property of approximately differentiable functions. (Russian). Mat. Zametki 41(4), 500–508 (1987)MathSciNetGoogle Scholar
  19. 19.
    Járai, A.: Derivatives are Borel functions. Aequationes Math. 29(1), 24–27 (1985).  https://doi.org/10.1007/BF02189809 MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kechris, A.S.: Classical Descriptive Set Theory, Volume 156 of Graduate Texts in Mathematics, vol. 156. Springer, New York (1995).  https://doi.org/10.1007/978-1-4612-4190-4 Google Scholar
  21. 21.
    Kelley, J.L.: General Topology. Springer, New York (1975). Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27zbMATHGoogle Scholar
  22. 22.
    Kolasiński, S., Menne, U.: Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds. NoDEA Nonlinear Differential Equations Appl. 24(2), Art. 17, 56 (2017).  https://doi.org/10.1007/s00030-017-0436-z
  23. 23.
    Liu, F.-C.: A localized Lusin theorem and a Rademacher type theorem. Bull. Inst. Math. Acad. Sin. (N.S.) 3(2), 243–253 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Lin, C.-L., Liu, F.-C.: Approximate differentiability according to Stepanoff-Whitney-Federer. Indiana Univ. Math. J. 62(3), 855–868 (2013).  https://doi.org/10.1512/iumj.2013.62.5024 MathSciNetzbMATHGoogle Scholar
  25. 25.
    Menne, U.: Some applications of the isoperimetric inequality for integral varifolds. Adv. Calc. Var. 2(3), 247–269 (2009).  https://doi.org/10.1515/ACV.2009.010 MathSciNetzbMATHGoogle Scholar
  26. 26.
    Menne, U.: A Sobolev Poincaré type inequality for integral varifolds. Calc. Var. Partial Differential Equations 38(3–4), 369–408 (2010).  https://doi.org/10.1007/s00526-009-0291-9 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Menne, U.: Decay estimates for the quadratic tilt-excess of integral varifolds. Arch. Ration. Mech. Anal. 204(1), 1–83 (2012).  https://doi.org/10.1007/s00205-011-0468-1 MathSciNetzbMATHGoogle Scholar
  28. 28.
    Menne, U.: A sharp lower bound on the mean curvature integral with critical power for integral varifolds, 2012. In: Abstracts from the Workshop Held July 22–28 (2012). Organized by Camillo De Lellis, Gerhard Huisken and Robert Jerrard, Oberwolfach Reports. Vol. 9, no. 3.  https://doi.org/10.4171/OWR/2012/36
  29. 29.
    Menne, U.: Second order rectifiability of integral varifolds of locally bounded first variation. J. Geom. Anal. 23(2), 709–763 (2013).  https://doi.org/10.1007/s12220-011-9261-5 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Menne, U.: Weakly differentiable functions on varifolds. Indiana Univ. Math. J. 65(3), 977–1088 (2016).  https://doi.org/10.1512/iumj.2016.65.5829 MathSciNetzbMATHGoogle Scholar
  31. 31.
    Menne, U.: Sobolev functions on varifolds. Proc. Lond. Math. Soc. (3) 113(6), 725–774 (2016).  https://doi.org/10.1112/plms/pdw023 MathSciNetzbMATHGoogle Scholar
  32. 32.
    Menne, U.: Pointwise differentiability of higher order for distributions (2018). arXiv:1803.10855v1
  33. 33.
    Michael, J.H., Simon, L.M.: Sobolev and mean-value inequalities on generalized submanifolds of \(R^{n}\). Comm. Pure Appl. Math. 26, 361–379 (1973).  https://doi.org/10.1002/cpa.3160260305 MathSciNetzbMATHGoogle Scholar
  34. 34.
    O’Neil, R.: Convolution operators and \(L(p,\, q)\) spaces. Duke Math. J. 30, 129–142 (1963)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Rešetnjak, J.G.: Generalized derivatives and differentiability almost everywhere. Math. USSR-Sb. 4, 293–302 (1968).  https://doi.org/10.1070/SM1968v004n03ABEH002799 Google Scholar
  36. 36.
    Reshetnyak, Y.G.: Generalized derivatives and differentiability almost everywhere. (Russian). Mat. Sb. (N.S.) 75(117), 323–334 (1968)MathSciNetGoogle Scholar
  37. 37.
    Santilli, M.: Rectifiability and approximate differentiability of higher order for sets. Indiana Univ. Math. J. (2017). To appear, available at arXiv:1701.07286v2
  38. 38.
    Schätzle, R.: Quadratic tilt-excess decay and strong maximum principle for varifolds. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5) 3(1), 171–231 (2004)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Schätzle, R.: Lower semicontinuity of the Willmore functional for currents. J. Differential Geom. 81(2), 437–456 (2009)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
  41. 41.
    Stein, E.M.: Editor’s note: the differentiability of functions in \({ R}^{n}\). Ann. of Math. (2) 113(2), 383–385 (1981)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Solomon, B., White, B.: A strong maximum principle for varifolds that are stationary with respect to even parametric elliptic functionals. Indiana Univ. Math. J. 38(3), 683–691 (1989).  https://doi.org/10.1512/iumj.1989.38.38032 MathSciNetzbMATHGoogle Scholar
  43. 43.
    Trudinger, N.S.: On the twice differentiability of viscosity solutions of nonlinear elliptic equations. Bull. Austral. Math. Soc. 39(3), 443–447 (1989).  https://doi.org/10.1017/S0004972700003361 MathSciNetzbMATHGoogle Scholar
  44. 44.
    Weil, C.E.: The Peano notion of higher order differentiation. Math. Japon. 42(3), 587–600 (1995)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Whitney, H.: On totally differentiable and smooth functions. Pacific J. Math. 1, 143–159 (1951)MathSciNetzbMATHGoogle Scholar
  46. 46.
    White, B.: The maximum principle for minimal varieties of arbitrary codimension. Comm. Anal. Geom. 18(3), 421–432 (2010).  https://doi.org/10.4310/CAG.2010.v18.n3.a1 MathSciNetzbMATHGoogle Scholar
  47. 47.
    Wickramasekera, N.: A general regularity theory for stable codimension 1 integral varifolds. Ann. of Math. (2) 179(3), 843–1007 (2014).  https://doi.org/10.4007/annals.2014.179.3.2 MathSciNetzbMATHGoogle Scholar
  48. 48.
    Wickramasekera, N.: A sharp strong maximum principle and a sharp unique continuation theorem for singular minimal hypersurfaces. Calc. Var. Partial Differential Equations 51(3–4), 799–812 (2014).  https://doi.org/10.1007/s00526-013-0695-4 MathSciNetzbMATHGoogle Scholar
  49. 49.
    Zibman, I.B.: Some characterizations of the \(n\)-dimensional Peano derivative. Studia Math. 63(1), 89–110 (1978).  https://doi.org/10.4064/sm-63-1-89-110 MathSciNetzbMATHGoogle Scholar
  50. 50.
    Ziemer, W.P.: Weakly Differentiable Functions, volume 120 of Graduate Texts in Mathematics. Springer, New York (1989). Sobolev spaces and functions of bounded variation.  https://doi.org/10.1007/978-1-4612-1015-3

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipei CityTaiwan, ROC

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