Skip to main content
SpringerLink
Log in

A Brief Overview on The Obstacle Problem

  • Conference paper
European Congress of Mathematics

Part of the book series: Progress in Mathematics ((PM,volume 202))

Abstract

We present a short survey on the obstacle problem including the theory developed by L. A. Caffarelli and the theory developed independently by G. S. Weiss. We also present some other recent results on the regularity of the free boundary.

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
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
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 AND A. FRIEDMANVariational Problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282(2), (1984), 431–461.

    Article  MathSciNet  Google Scholar 

  2. I. BLANK, Sharp Results for the Regularity and Stability of the Free Boundary in the Obstacle Problem, to appear in Indiana Univ. Math. J.

    Google Scholar 

  3. H. Brezis and D. Kinderlehrer, The Smoothness of Solutions to Nonlinear Variational Inequalities,Indiana Univ. Math. J. 23(9), (1974), 831–844.

    MathSciNet  MATH  Google Scholar 

  4. L. A. Caffarelli, Free boundary problem in highter dimensions, Acta Math. 139, (1977), 155–184.

    Article  MathSciNet  Google Scholar 

  5. L. A. CAFFARELLI, Compactness Methods in Free Boundary Problems,Comm. Partial Differential Equations 5(4), (1980), 427–448.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. A. CAFFARELLI, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets, Boll. Un. Mat. Ital. A 18(5), (1981), 109–113.

    MathSciNet  MATH  Google Scholar 

  7. L. A. CAFFARELLI, The Obstacle Problem revisited, J. Fourier Anal. Appl. 4, (1998), 383–402.

    Article  MathSciNet  MATH  Google Scholar 

  8. L. A. CAFFARELLI AND N.M. RIVIERESmoothness and Analyticity of Free Boundaries in Variational Inequalities, Ann. Scuola Norm. Sup. Pisa, serie IV 3, (1975), 289–310.

    Google Scholar 

  9. L. A. CAFFARELLI AND N. M. RIVIEREAsymptotic behaviour of free boundaries at their singular points, Ann. of Math. 106, (1977), 309–317.

    Article  MathSciNet  Google Scholar 

  10. [10] J. DOLBEAULT AND R. MONNEAU, Convexity Properties of the Free Boundary and Gradient Estimates for Quasi-linear Elliptic Equations, to appear.

    Google Scholar 

  11. J. FREHSEOn the Regularity of the Solution of a Second Order Variational Inequality, Boll. Un. Mat. Ital. B (7) 6(4), (1972), 312–315.

    MathSciNet  MATH  Google Scholar 

  12. A. FRIEDMAN, Variational Principles and Free Boundary Problems, Pure and applied mathematics, ISSN 0079–8185, a Wiley-Interscience publication, (1982).

    Google Scholar 

  13. A. FRIEDMAN AND D. PHILLIPS, The free boundary of a semilinear elliptic equation, Trans. Amer. Math. Soc. 282, (1984), 153–182.

    Article  MathSciNet  MATH  Google Scholar 

  14. V. ISAKOV, Inverse theorems on the smoothness of potentials, Differential Equations 11, (1976), 50–57.

    MATH  Google Scholar 

  15. B. KAwOHL, When are solutions to nonlinear elliptic boundary value problems convex?, Comm. Partial Differential Equations 10, (1985), 1213–1225.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. KINDERLEHRER AND L. NIRENBERG, Regularity in free boundary problems, Ann. Scuola Norm. Sup. Pisa 4, (1977), 373–391.

    MathSciNet  MATH  Google Scholar 

  17. D. KINDERLEHRER AND G. STAMPACCHIA, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, (1980).

    MATH  Google Scholar 

  18. R. MONNEAU, Problèmes de frontières libres, EDP elliptiques non linEaires et applications en combustion, supraconductivitE et ElasticitE, Doctoral Dissertation, UniversitE Pierre et Marie Curie, Paris (1999).

    Google Scholar 

  19. R. MONNEAU, On the Number of Singularities for the Obstacle Problem in Two Dimensions, to appear.

    Google Scholar 

  20. J. F. RODRIGUES, Obstacle Problems in Mathematical Physics,North-Holland, (1987).

    MATH  Google Scholar 

  21. D. G. SCHAEFFERAn Example of Generic Regularity for a Non-Linear Elliptic Equation, Arch. Rat. Mach. Anal. 57, (1974), 134–141.

    MathSciNet  MATH  Google Scholar 

  22. D. G. SCHAEFFER, A Stability Theorem for the Obstacle problem, Advances in Math. 16, (1975), 34–47.

    Article  Google Scholar 

  23. D. G. SCHAEFFER, Some Examples of Singularities in a Free Boundary, Ann. Scuola Norm. Sup. Pisa 4(4), (1976), 131–144.

    Google Scholar 

  24. D. G. SCHAEFFEROne-sided Estimates for the Curvature of the Free Boundary in the Obstacle Problem, Advances in Math. 24, (1977), 78–98.

    Article  MathSciNet  MATH  Google Scholar 

  25. G. S. WEISS, A homogeneity improvement approach to the obstacle problem, Invent. Math. 138, (1999), 23–50.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ecole Nationale des Ponts et Chaussées, CERMICS, 6 et 8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, Marne-la-Vallée Cedex 2, 77455, France

    Regis Monneau

Authors
  1. Regis Monneau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    Carles Casacuberta  & Joan Verdera  & 

  2. Departament d’Algebra i Geometria Facultat de Matemàtiques, Universitat de Barcelona, 08007, Barcelona, Spain

    Rosa Maria Miró-Roig

  3. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, 08028, Barcelona, Spain

    Sebastià Xambó-Descamps

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this paper

Cite this paper

Monneau, R. (2001). A Brief Overview on The Obstacle Problem. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8266-8_26

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9496-8

  • Online ISBN: 978-3-0348-8266-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation