Advertisement

Chinese Annals of Mathematics, Series B

, Volume 39, Issue 5, pp 917–932 | Cite as

Measure Estimates of Nodal Sets of Polyharmonic Functions

  • Long Tian
Article
  • 2 Downloads

Abstract

This paper deals with the function u which satisfies △ku = 0, where k ≥ 2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u.

Keywords

Polyharmonic function Nodal set Frequency Measure estimate Growth property 

2000 MR Subject Classification

58E10 53C50 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Almgren, F. J., Dirichlet’s problem for muptiple valued functions and the regularity of Mass minimizing integral currents, Minimal Submanifolds and Geodesics, Obata, M. (ed.), North Holland, Amsterdam, 1979, 1–6.Google Scholar
  2. [2]
    Donnelly, H. and Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93, (1988), 161–183.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Federer, H., Geometric Measure Theory, Springer-Verlag, New York, 1969.zbMATHGoogle Scholar
  4. [4]
    Garofalo, N. and Lin, F. H., Monotonicity properties of variational integrals, Ap-weights and unique continuation, Indiana Univ. Math. J., 35, (1986), 245–267.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Garofalo, N. and Lin, F. H., Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure. Appl. Math., 40, (1987), 347–366.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Han, Q., Schäuder estimates for elliptic operators with applications to nodal sets, The Journal of Geometric Analysis, 10(3), (2000), 455–480.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Han, Q., Hardt, R. and Lin, F. H., Singular sets of higher order elliptic equations, Communications in Partial Differential Equations, 28, (2003), 2045–2063.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Han, Q. and Lin, F. H., Nodal sets of solutions of elliptic differential equations, http://nd.edu/qhan/nodal.pdf.
  9. [9]
    Lin, F. H., Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure. Appl. Math., 44, (1991), 287–308.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Lin, F. H. and Yang, X. P., Geometric Measure Theory: An Introduction, Advanced Mathematics, 1, Science Press, Beijing; International Press, Boston, 2002.Google Scholar
  11. [11]
    Liu, H. R., Tian, L. and Yang, X. P., The growth of H-harmonic functions on the Heisenberg group, Science China Mathematics, 54(4), (2014), 795–806.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Schoen, R. M. and Yau, S. T., Lectures on Differential Geometry, International Press, Cambridge, MA, 1994.zbMATHGoogle Scholar
  13. [13]
    Tian, L. and Yang, X. P., Measure estimates of nodal sets of bi-harmonic functions, Journal of Differential Equations, 256(2), (2014), 558–576.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied MathematicsNanjing University of Science and TechnologyNanjingChina

Personalised recommendations