Skip to main content
SpringerLink
Log in

Boundary estimates for solutions of two-phase obstacle problems

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We establish the optimal regularity (of class W 2∞ ) of a solution to the two-phase obstacle problem

$$\Delta u = \lambda ^ + \chi _{\{ u > 0\} } - \lambda ^ - \chi _{\{ u < 0\} } in \Omega , \lambda ^ \pm \geqslant 0, \lambda ^ + + \lambda ^ - > 0,$$

with a nonhomogeneous Dirichlet condition in a bounded domain Ω ⊂ ℝn with smooth boundary ∂Ω. Bibliography: 10 titles.

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. G. S. Weiss, “An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary,” Interfaces Free Bound. 3 (2001), 121–128.

    Article  MATH  MathSciNet  Google Scholar 

  2. R. Jensen, “Boundary regularity for variational inequalities,” Indiana Univ. Math. J. 29 (1980), no. 4, 495–504.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. N. Uraltseva, “Two-phase obstacle problem” [in Russian], Probl. Mat. Anal. 22 (2001), 240–245; English transl.: J. Math. Sci. (New York) 106 (2001), 3073–3077.

    MathSciNet  Google Scholar 

  4. H. Shahgholian, “C 1,1-regularity in semilinear elliptic problems,” Comm. Pure Appl. Math. 56 (2003), 278–281.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Andersson, N. Matevosyan, and H. Mikayelyan, “On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem,” Ark. Mat. 44 (2006), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  6. N. N. Uraltseva, “Boundary estimates for solutions of elliptic and parabolic equations with discontinuous nonlinearities,” Am. Math. Soc. Transl. (2). [To appear]

  7. H. W. Alt, L. A. Caffarelli, and A. Friedman, “Variational problems with two phases and their free boundaries,” Trans. Amer. Math. Soc. 282 (1984), no. 2, 431–461.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    D. E. Apushkinskaya

  2. St.-Petersburg State University, Russia

    N. N. Uraltseva

Authors
  1. D. E. Apushkinskaya
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. N. Uraltseva
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

__________

Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 3–11.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Apushkinskaya, D.E., Uraltseva, N.N. Boundary estimates for solutions of two-phase obstacle problems. J Math Sci 142, 1723–1732 (2007). https://doi.org/10.1007/s10958-007-0083-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-007-0083-8

Keywords

Search

Navigation