Skip to main content
SpringerLink
Log in

Schauder estimates for elliptic operators with applications to nodal sets

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

In this paper we first give a priori estimates on asymptotic polynomials of solutions to elliptic equations at nodal points. This leads to a pointwise version of Schauder estimates. As an application we discuss the structure of nodal sets of solutions to elliptic equations with nonsmooth coefficients.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,Comm. Pure Appl. Math.,12, 623–727, (1959).

    Article  MathSciNet  MATH  Google Scholar 

  2. Agmon, S., Douglis, A., and Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II,Comm. Pure Appl. Math.,17, 35–92, (1964).

    Article  MathSciNet  MATH  Google Scholar 

  3. Bers, L. Local behavior of solution of general linear elliptic equations,Comm. Pure Appl. Math.,8, 473–496, (1955).

    Article  MathSciNet  MATH  Google Scholar 

  4. Caffarelli, L.A. Interiora priori estimates for solutions of fully nonlinear equations,Ann. Math.,130, 189–213, (1989).

    Article  MathSciNet  Google Scholar 

  5. Caffarelli, L.A. Elliptic second order equations, Rendiconti del Seminario Matematico e Fisico di Milano, LVIII, 253–284, 1988.

  6. Caffarelli, L.A. and Friedman, A. Partial regularity of the zero-set of solutions of linear and superlinear elliptic equations,J. Diff. Eq.,60, 420–433, (1985).

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L.A. and Kenig, C. Gradient estimates for variable coefficient parabolic equations and singular perturbation problem,Amer. J. Math.,120, 391–439, (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. Dong, R.-T. Nodal sets of eigenfunctions on Riemann surfaces,J. Diff. Geom.,36, 493–506, (1992).

    MATH  Google Scholar 

  9. Donnelly, H. and Fefferman, C. Nodal sets for eigenfunctions of the Laplacian on surfaces,J. Am. Math. Soc., to appear.

  10. Federer, H.Geometric Measure Theory, Springer-Verlag, New York, 1969.

    MATH  Google Scholar 

  11. Garofalo, N. and Lin, F.-H. Monotonicity properties of variational integrals,A P weights and unique continuation,Indiana Univ. Math. J.,35, 245–267, (1986).

    Article  MathSciNet  MATH  Google Scholar 

  12. Gilbarg, D. and Trudinger, N.Elliptic Partial Differential Equations of Second Order, 2 Ed., Springer-Verlag, Berlin, 1983.

    MATH  Google Scholar 

  13. Han, Q. Singular sets of solutions to elliptic equations,Indiana Univ. Math. J.,43, 983–1002, (1994).

    Article  MathSciNet  MATH  Google Scholar 

  14. Han, Q. and Lin, F.-H. Nodal sets of solutions of parabolic equations, II,Comm. Pure Appl. Math., 1219–1238, (1994).

  15. Hardt, R. and Simon, L. Nodal sets for solutions of elliptic equations,J. Diff. Geom.,30, 505–522, (1989).

    MathSciNet  MATH  Google Scholar 

  16. Hoffman-Ostenhof, M., Hoffman-Ostenhof, T., and Nadirashvili, N. Interior Holder estimates for solutions of Schrodinger equations and regularity of nodal sets,Comm. Partial Diff. Eq.,20, 1241–1273, (1995).

    Article  Google Scholar 

  17. Hoffman-Ostenhof, M., Hoffman-Ostenhof, T., and Nadirashvili, N. Interior Holder estimates for solutions of Schrodinger equations and regularity ofnodal sets, Journees “Equations aux Derivees Partielles” (Saint-Jean-de-Monts, 1994), Exp. No. XIII, 9pp., Ecole Polytech., Palaiseau, 1994.

    Google Scholar 

  18. Hoffman-Ostenhof, M., Hoffman-Ostenhof, T., and Nadirashvili, N. Critical sets of smooth solutions to elliptic equations in dimension 3,Indiana Univ. Math. J.,45, 15–37, (1996).

    Article  MathSciNet  Google Scholar 

  19. Lin, F.-H. Nodal sets of solutions of elliptic and parabolic equations,Comm. Pure Appl. Math.,45, 287–308, (1991).

    Article  Google Scholar 

  20. Simon, L.Lectures on Geometric Measure Theory, Proc. C.M.A., Aust. National University, 1983.

  21. Simon, L.Schauder Estimates by Scaling, to appear.

  22. Stein, M.Singular Integrals and the Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

    Google Scholar 

  23. Troianiello, G.M.Differential Equations and Obstacle Problem, Plenum Press, New York, 1987.

    Google Scholar 

  24. Trudinger, N.S. A new approach to the Schauder estimates for linear elliptic equation,Proc. C.M.A., Aust. Nat. Univ., 52–59, 1986.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, 46556, Notre Dame, IN

    Qing Han

Authors
  1. Qing Han
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing Han.

Additional information

Communicated by David Jenson

Rights and permissions

Reprints and permissions

About this article

Cite this article

Han, Q. Schauder estimates for elliptic operators with applications to nodal sets. J Geom Anal 10, 455–480 (2000). https://doi.org/10.1007/BF02921945

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02921945

Math Subject Classifications

Key Words and Phrases

Search

Navigation