Skip to main content
SpringerLink
Log in

Existence of solutions of Two-Phase free boundary problems for fully nonlinear elliptic equations of second order

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

Abstract

In this article, we have established existence of a solution to the 2 -phase free boundary problem for some fully nonlinear elliptic equations and also shown the free boundary has finite Hn−1 Hausdorff measure and a normal in a measuretheoretic sense Hn−1 almost everywhere. The regularity theory developed in [9] and [10] for this free boundary problem then leads to the fact that the free boundary is locally a C1,α surface near Hn−1-a.e. point.

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. Alt, H.W., Caffarelli, L.A., and Friedman, A. Variational problems with two phases and their free boundaries,T.A.M.S.,282(2), 431–461, (1984).

    Article  MathSciNet  Google Scholar 

  2. Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part 1: Lipschitz free boundaries are C1,α,Revista Matematica Ibewamericana,3(2), 139–162, (1987).

    MathSciNet  MATH  Google Scholar 

  3. Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part 2: Flat free boundaries are Lipschitz,Comm. Pure Appl. Math,42(1), 55–78, (1989).

    Article  MathSciNet  MATH  Google Scholar 

  4. Caffarelli, L.A. A Harnack inequality approach to the regularity of free boundaries, Part 3: Existence theory, compactness, and dependence on X,Ann. Scuola Norm. Sup. Piza Cl. Sci,15(4), 583–602, (1988).

    MathSciNet  MATH  Google Scholar 

  5. Cabré, X. and Caffarelli, L.A. Fully nonlinear elliptic equations,Am. Math. Soc., (1995).

  6. Evens, L.C. and Gariepy, R.F.Measure Theory and Fine Properties of Functions, CRC Press, Inc., (1992).

  7. Gilbarg, D. and Trudinger N.S.Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, (1983).

  8. Miller, K. Barriers on cones for uniformly elliptic perators,Ann. Mat. Pura Appl.,76, 93–105, (1967).

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, P. Regularity of Free Boundaries of Two-Phase Problems for fully nonlinear elliptic equations of second order, Part 1: Lipschitz free boundaries are C1,α,Comm Pure Appl. Math.,53(7), 799–810, (2000).

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, P. Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. part 2: Flat free boundaries are Lipschitz,Comm in P.D.E.,27(7/8), 1497–1514, (2002).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, 93106, Santa Barbara, CA

    Pei-Yong Wang

Authors
  1. Pei-Yong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, PY. Existence of solutions of Two-Phase free boundary problems for fully nonlinear elliptic equations of second order. J Geom Anal 13, 715–738 (2003). https://doi.org/10.1007/BF02921886

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02921886

Math Subject Classifications

Key Words and Phrases

Search

Navigation