Skip to main content
SpringerLink
Log in

Stability remarks to the obstacle problem for p-Laplacian type equations

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract.

We extend some convergence and L 1 stability results for the coincidence set to the p-obstacle problem under natural nondegeneracy conditions and without restrictions on p, \(1\! < \!p\! < \!\infty\). We rely on the local \(C^{1,\lambda}\) regularity of the solution and, as an application, we show the existence of a solution to the thermal membrane problem, and in a limit nonlocal case also its uniqueness for small data.

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. Brézis, H.: Nouveaux théoréms de régularité pour les problémes unilatéraux. Rencontre entre Physicien Théoricien et Mathématiciens, Strasbourg 12 (1971)

  2. Brézis, H., Strauss, W.: Semi-linear second-order elliptic equations in L 1. J. Math. Soc. Japan 25, 565-590 (1973)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  5. Choe, H.J.: A regularity theory for a general class of quasilinear elliptic partial differential equations and obstacle problems. Arch. Rational Mech. Anal. 114, 383-394 (1991)

    Article  Google Scholar 

  6. Choe, H.J., Lewis, J.: On the obstacle problem for quasilinear elliptic equations of p Laplacian type. SIAM J. Math. Anal. 22, 623-638 (1991)

    Article  Google Scholar 

  7. DiBenedetto, E.: \(C^{1+\alpha}\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7, 827-850 (1983)

    Article  Google Scholar 

  8. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Reprint of the 1998 edn., Classics in Mathematics. Springer, Berlin 2001

  9. Karp, L., Kilpeläinen, T., Petrosyan, A., Shahgholian, H.: On the porosity of free boundaries in degenerate variational inequalities. J. Differential Equations 164, 110-117 (2000)

    Article  Google Scholar 

  10. Lee, K., Shahgholian, H.: Hausdorff dimension and stability for the p-obstacle problem (\(2<p<\infty\)). J. Differential Equations 195, 14-24 (2003)

    Article  Google Scholar 

  11. Lewy, H., Stampacchia, G.: On the smoothness of superharmonics which solve a minimum problem. J. Analyse Math. 23, 224-236 (1970)

    Google Scholar 

  12. Fuchs, M.: Hölder continuity of the gradient for degenerate variational inequalities. Nonlinear Anal. 15, 85-100 (1990)

    Article  Google Scholar 

  13. Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Academic Press, New York London 1980

  14. Mu, J., Ziemer, W.P.: Smooth regularity of solutions of double obstacle problems involving degenerate elliptic equations, Comm. Partial Differential Equations 16, 821-843 (1991)

    Google Scholar 

  15. Mu, J.: Higher regularity of the solution to the p-Laplacian obstacle problem. J. Differential Equations 95, 370-384 (1992)

    Article  Google Scholar 

  16. Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Advances in Math. 3, 510-585 (1969)

    Article  Google Scholar 

  17. Frehse, J., Mosco, U.: Irregular obstacles and quasivariational inequalities of stochastic impulse control. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 9, 105-157 (1982)

    Google Scholar 

  18. Ladyzhenskaya, O., Ural’tseva, N.: Linear and quasilinear elliptic equations. Academic Press, New York London 1968

  19. Léné, F.: Sur les matériaux élastiques á énergie de déformation non quadratique. J. Mécanique 13, 499-534 (1974)

    Google Scholar 

  20. Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203-1219 (1988)

    Article  Google Scholar 

  21. Lieberman, G.: Regularity of solutions to some degenerate double obstacle problems. Indiana Univ. Math. J. 40, 1009-1028 (1991)

    Article  Google Scholar 

  22. Lindqvist, P.: Regularity for the gradient of the solution to a nonlinear obstacle problem with degenerate ellipticity. Nonlinear Anal. 12, 1245-1255 (1988)

    Article  Google Scholar 

  23. Lions, J.-L.: Quelques méthodes de résolution des problémes aux limites non linéaires (French) Dunod. Gauthier-Villars, Paris 1969

  24. Norando, T.: \(C^{1,\alpha}\) local regularity for a class of quasilinear elliptic variational inequalities. Boll. Un. Mat. Ital. C (6) 5, 281-292 (1986)

  25. Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51, 126-150 (1984)

    Article  Google Scholar 

  26. Rodrigues, J.F.: On the stability of the free boundary in the obstacle problem for a minimal surface. Portugaliae Math. 41, 463-480 (1982)

    Google Scholar 

  27. Rodrigues, J.F.: Obstacle problems in mathematical physics. North-Holland, Amsterdam 1987

  28. Rodrigues, J.F.: On a new class of nonlocal unilateral problems in thermomechanics, Lectures on applied mathematics (Munich, 1999). Springer, Berlin 2000, pp. 89-101

Download references

Author information

Authors and Affiliations

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

    José Francisco Rodrigues

Authors
  1. José Francisco Rodrigues
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Francisco Rodrigues.

Additional information

Received: 2 October 2003, Accepted: 10 May 2004, Published online: 10 December 2004

Mathematics Subject Classification (2000):

35J85, 49J40, 35R35, 47J20

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rodrigues, J.F. Stability remarks to the obstacle problem for p-Laplacian type equations. Calc. Var. 23, 51–65 (2005). https://doi.org/10.1007/s00526-004-0288-3

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-004-0288-3

Keywords

Search

Navigation