Decay Estimates for the Quadratic Tilt-Excess of Integral Varifolds

  • Ulrich MenneEmail author


This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space with their first variation given by either a Radon measure or a function in some Lebesgue space. Pointwise decay results for the quadratic tiltexcess are established for those varifolds. The results are optimal in terms of the dimension of the varifold and the exponent of the Lebesgue space in most cases, for example if the varifold is not two-dimensional.


Radon Radon Measure Lebesgue Space Decay Estimate Partial Regularity 
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  1. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)Google Scholar
  2. Allard W.K.: On the first variation of a varifold. Ann. Math. 2(95), 417–491 (1972)MathSciNetCrossRefGoogle Scholar
  3. Almgren F.J. Jr.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Ann. Math. 2(87), 321–391 (1968)MathSciNetCrossRefGoogle Scholar
  4. Almgren F.: Optimal isoperimetric inequalities. Indiana Univ. Math. J. 35(3), 451–547 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Almgren, F.J. Jr.: Almgren’s big regularity paper. World Scientific Monograph Series in Mathematics, vol. 1. 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 SchefferGoogle Scholar
  6. Bombieri E., De Giorgi E., Giusti E.: Minimal cones and the Bernstein problem. Invent. Math. 7, 243–268 (1969)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. Brakke K.A.: The Motion of a Surface by its Mean Curvature Mathematical Notes, vol. 20. Princeton University Press, Princeton (1978)Google Scholar
  8. Caffarelli L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math.(2) 130(1), 189–213 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  9. Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics, vol 580. Springer (1977)zbMATHGoogle Scholar
  10. De Giorgi E.: Frontiere orientate di misura minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–61. Editrice Tecnico Scientifica, Pisa (1961)Google Scholar
  11. De Giorgi, E.: Selected Papers. Ambrosio, L., Dal Maso, G., Forti, M., Miranda, M., Spagnolo, S. (eds.) Springer, Berlin 2006Google Scholar
  12. Duzaar F., Mingione G.: Harmonic type approximation lemmas. J. Math. Anal. Appl. 352(1), 301–335 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Duzaar F., Steffen K.: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546, 73–138 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. Federer H.: Geometric Measure Theory Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969)Google Scholar
  15. Federer H.: Flat chains with positive densities. Indiana Univ. Math. J. 35(2), 413–424 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Giusti E.: Direct Methods in the Calculus of Variations. World Scientific Publishing Co. Inc., River Edge (2003)zbMATHCrossRefGoogle Scholar
  17. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer-Verlag, Berlin, 2001. Reprint of the 1998 editionGoogle Scholar
  18. Hutchinson J.: Poincaré-Sobolev and related inequalities for submanifolds of R N. Pacific J. Math. 145(1), 59–69 (1990)MathSciNetzbMATHGoogle Scholar
  19. Jin T., Maz’ya V., Van Schaftingen J.: Pathological solutions to elliptic problems in divergence form with continuous coefficients. C. R. Math. Acad. Sci. Paris 347(13–14), 773–778 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. Kelley, J.L.: General Topology. Springer, New York, 1975. Reprint of the 1955 edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27Google Scholar
  21. Leonardi G.P., Masnou S.: Locality of the mean curvature of rectifiable varifolds. Adv. Calc. Var. 2(1), 17–42 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  22. Menne, U.: \({\mathcal{C}^2}\) Rectifiability and Q Valued Functions. PhD thesis, Universität Tübingen, 2008.
  23. Menne U.: Some applications of the isoperimetric inequality for integral varifolds. Adv. Calc. Var. 2(3), 247–269 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  24. Menne U.: A Sobolev Poincaré type inequality for integral varifolds. Calc. Var. Partial Differ. Equ. 38(3–4), 369–408 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. Menne, U.: Second order rectifiability of integral varifolds of locally bounded first variation. J. Geom. Anal. (2011). doi: 10.1007/s12220-011-9261-5
  26. Mingione G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  27. O’Neil R.: Convolution operators and L(p, q) spaces. Duke Math. J. 30, 129–142 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Schätzle R.: Hypersurfaces with mean curvature given by an ambient Sobolev function. J. Differ. Geom. 58(3), 371–420 (2001)MathSciNetzbMATHGoogle Scholar
  29. 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
  30. Schätzle R.: Lower semicontinuity of the Willmore functional for currents. J. Differ. Geom. 81(2), 437–456 (2009)zbMATHGoogle Scholar
  31. Simon, L.: Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, vol. 3. Australian National University. Australian National University Centre for Mathematical Analysis, Canberra, 1983Google Scholar
  32. Simon L.: Schauder estimates by scaling. Calc. Var. Partial Differ. Equ. 5(5), 391–407 (1997)zbMATHCrossRefGoogle Scholar
  33. Schoen R., Simon L.: A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J. 31(3), 415–434 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  34. Trudinger N.S.: On the twice differentiability of viscosity solutions of nonlinear elliptic equations. Bull. Austral. Math. Soc. 39(3), 443–447 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  35. Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge, 1987. Translated from the German by C.B. Thomas and M.J. ThomasGoogle Scholar

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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Max-Planck-Institute for Gravitational Physics (Albert-Einstein-Institute)PotsdamGermany

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