Skip to main content
Log in

Γ-Convergence of some super quadratic functionals with singular weights

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

We study the Γ-convergence of the following functional (p > 2)

$$F_{\varepsilon}(u):=\varepsilon^{p-2}\int\limits_{\Omega} |Du|^p d(x,\partial \Omega)^{a}dx+\frac{1}{\varepsilon^{\frac{p-2}{p-1}}} \int\limits_{\Omega} W(u) d(x,\partial \Omega)^{-\frac{a}{p-1}}dx+\frac{1}{\sqrt{\varepsilon}} \int\limits_{\partial\Omega} V(Tu)d\mathcal{H}^2,$$

where Ω is an open bounded set of \({\mathbb{R}^3}\) and W and V are two non-negative continuous functions vanishing at α, β and α′, β′, respectively. In the previous functional, we fix a = 2 − p and u is a scalar density function, Tu denotes its trace on ∂Ω, d(x, ∂Ω) stands for the distance function to the boundary ∂Ω. We show that the singular limit of the energies \({F_{\varepsilon}}\) leads to a coupled problem of bulk and surface phase transitions.

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

Access this article

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. Adams R., Fournier J.J.F.: Sobolev Spaces, 2nd edn. Academic Press, Oxford (2003)

    MATH  Google Scholar 

  2. Alberti G., Bouchitté G., Seppecher P.: Un résultat de perturbations singuliéres avec la norme H 1/2. C. R. Acad. Sci. Paris, Série I 319, 333–338 (1994)

    MATH  Google Scholar 

  3. Alberti G., Bouchitté G., Seppecher P.: Phase transition with line-tension effect. Arch. Rational Mech. Anal. 144, 1–46 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, Oxford (2000)

    MATH  Google Scholar 

  5. Berestycki H., Lachand-Robert T.: Some properties of monotone rearrangement with applications to elliptic equations in cylinders. Math. Nachr. 266, 3–19 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Caffarelli L., Silvestre L.: An extension problem related to the fractional laplacian. Commun. PDE 32, 1245–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Evans L.C., Gariepy R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992)

    MATH  Google Scholar 

  8. Garroni A., Palatucci G.: A singular perturbation result with a fractional norm. In: Dal Maso, G., De Simone, A., Tomarelli, F. (eds) Variational Problems in Material Science. Progress in NonLinear Differential Equations and Their Applications, vol. 68, pp. 111–126. Birkhäuser, Basel (2006)

    Google Scholar 

  9. Gonzalez, M.D.M.: Γ-convergence of an energy functional related to the fractional Laplacian. Calc. Var. PDE (2009) (to appear)

  10. Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser, Basel (1984)

    MATH  Google Scholar 

  11. Modica L.: Gradient theory of phase transitions and minimal interface criterion. Arch. Rational Mech. Anal. 98, 123–142 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Modica L.: Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 453–486 (1987)

    Google Scholar 

  13. Modica L., Mortola S.: Un esempio di Γ-convergenza. Boll. Un. Mat. Ital. 14(B5), 285–299 (1977)

    MATH  MathSciNet  Google Scholar 

  14. Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    MATH  MathSciNet  Google Scholar 

  15. Nekvinda A.: Characterization of traces of the weighted Sobolev space \({W^{1,p}(\Omega, d^{\varepsilon}_{M})}\) on M. Czechoslovak Math. J. 43(4), 695–711 (1993)

    MATH  MathSciNet  Google Scholar 

  16. Palatucci, G.: A class of phase transitions problems with the line tension effect. Ph.D. Thesis (2007). Avalaible online at http://cvgmt.sns.it/people/palatucci/

  17. Palatucci G.: Phase transitions with line tension: the super-quadratic case. Math. Models Methods Appl. Sci. 19(10), 1–31 (2009)

    MathSciNet  Google Scholar 

  18. Valadier M.: Young measures in methods of nonconvex analysis. Lecture Notes in Mathematics, vol. 1446, pp. 152–188. Springer-Verlag, New York (1990)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yannick Sire.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Palatucci, G., Sire, Y. Γ-Convergence of some super quadratic functionals with singular weights. Math. Z. 266, 533–560 (2010). https://doi.org/10.1007/s00209-009-0584-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-009-0584-x

Keywords

Mathematics Subject Classification (2000)

Navigation