Skip to main content
SpringerLink
Log in

Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries

  • Conference paper
  • First Online:
Free Boundary Problems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 154))

  • 1549 Accesses

Abstract

The first part of this paper is an announcement of a result to appear. We apply the Aleksandrov reflection to obtain regularity and stability of the free boundaries in the two-dimensional problem

$$ \Delta u = \frac{{\lambda _ + }} {2}\chi \{ u > 0\} - \frac{{\lambda _ - }} {2}\chi \{ u < 0\} ,$$

where λ+ > 0 and γ− > 0.

In the second part we show that the Kelvin reflection can be used in a similar way to obtain regularity of the classical obstacle problem

$$ \Delta u = \chi \{ u > 0\}$$

in higher dimensions.

H. Shahgholian has been partially supported by the Swedish Research Council. G.S. Weiss has been partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan. He wishes to thank the Max Planck Institute for Mathematics in the Sciences for the hospitality during his stay in Leipzig and the Göran Gustafsson Foundation for visiting appointments to the Royal Inst. of Technology in Stockholm.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (1984), 431–461.

    Article  MathSciNet  Google Scholar 

  2. M.-F. Bidaut-Véron, V. Galaktionov, P. Grillot, L. Véron, Singularities for a 2-dimensional semilinear elliptic equation with a non-Lipschitz nonlinearity. J. Differential Equations 154 (1999), 318–338.

    Article  MathSciNet  Google Scholar 

  3. H. Brézis, D. Kinderlehrer, The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23 (1973/74), 831–844.

    Article  MathSciNet  Google Scholar 

  4. L.A. Caffarelli, The regularity of free boundaries in higher dimensions. Acta Math. 139 (1977), 155–184.

    Article  MathSciNet  Google Scholar 

  5. L.A. Caffarelli, L. Karp, H. Shahgholian, Regularity of a free boundary in potential theory with application to the Pompeiu problem, Ann. of Math. (2) 151 (2000), 269–292.

    Article  MathSciNet  Google Scholar 

  6. X.-Y. Chen, Uniqueness of the ω-limit point of solutions of a semilinear heat equation on the circle. Proc. Japan Acad. Ser. A Math. Sci. 62 (1986), 335–337.

    Article  MathSciNet  Google Scholar 

  7. G. Duvaut, J.-L. Lions, Inequalities in mechanics and physics. Translated from the French by C. W. John. Grundlehren der Mathematischen Wissenschaften, 219. Springer-Verlag, Berlin-New York, 1976.

    Book  Google Scholar 

  8. J. Frehse, On the regularity of the solution of a second order variational inequality, Boll. Un. Mat. Ital. (4) 6 (1972), 312–315.

    MathSciNet  MATH  Google Scholar 

  9. X.-Y. Chen, H. Matano, L. Véron, Anisotropic singularities of solutions of nonlinear elliptic equations in R 2. J. Funct. Anal. 83 (1989), 50–97.

    Article  MathSciNet  Google Scholar 

  10. B. Chow, R. Gulliver, Aleksandrov reflection and geometric evolution of hypersurfaces. Comm. Anal. Geom. 9 (2001), 261–280.

    Article  MathSciNet  Google Scholar 

  11. B. Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations. Comm. Anal. Geom. 5 (1997), 389–409.

    Article  MathSciNet  Google Scholar 

  12. B. Chow, R. Gulliver, Aleksandrov reflection and nonlinear evolution equations. I. The n-sphere and n-ball. Calc. Var. Partial Differential Equations 4 (1996), 249–264.

    Article  MathSciNet  Google Scholar 

  13. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order. Second edition. Springer-Verlag, Berlin, 1983

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. H. Shahgholian, N.N. Uraltseva, G.S. Weiss, Global solutions of an obstacle-problem-like equation with two phases. Monatsh. Math. 142 (2004), 27–34.

    Article  MathSciNet  Google Scholar 

  16. N.N. Uraltseva, Two-phase obstacle problem. Problems in Math.Analysis, v. 22, 2001, 240–245 (in Russian. English translation: Journal of Math. Sciences, v.106, N 3, 2001, pp. 3073–3078).

    Google Scholar 

  17. G.S. Weiss, A Homogeneity Improvement Approach to the Obstacle Problem, Inv. Math. 138 (1999), 23–50.

    Article  MathSciNet  Google Scholar 

  18. G.S. Weiss, Partial Regularity for Weak Solutions of an Elliptic Free Boundary Problem, Commun. Partial Differ. Equations 23 (1998), 439–455.

    Article  MathSciNet  Google Scholar 

  19. 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 and Free Boundaries 3 (2001), 121–128.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Royal Institute of Technology, S-100 44, Stockholm, Sweden

    Henrik Shahgholian

  2. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo-to, 153-8914, Japan

    Georg S. Weiss

  3. Guest of the Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103, Leipzig, Germany

    Georg S. Weiss

Authors
  1. Henrik Shahgholian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georg S. Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Apartado 3008, 3001-454, Coimbra, Portugal

    Isabel Narra Figueiredo

  2. Universidade de Lisboa / CMAF, Av. Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    José Francisco Rodrigues

  3. Departamento de Matemática, Universidade do Minho, Campus de Gualtar, 4710-057, Braga, Portugal

    Lisa Santos

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Shahgholian, H., Weiss, G.S. (2006). Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries. In: Figueiredo, I.N., Rodrigues, J.F., Santos, L. (eds) Free Boundary Problems. International Series of Numerical Mathematics, vol 154. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7719-9_38

Download citation

Publish with us

Policies and ethics

Search

Navigation