Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

New monotonicity formulae for semi-linear elliptic and parabolic systems

  • 98 Accesses

  • 2 Citations

Abstract

The authors establish a general monotonicity formula for the following elliptic system

$$ \Delta u_i + f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \Omega $$

where Ω ⊂⊂ ℝn is a regular domain, (f i (x, u 1, ..., u m )) = ∇\( \vec u \) F(x, \( \vec u \)), F(x, \( \vec u \)) is a given smooth function of x ∈ ℝn and \( \vec u \) = (u 1, ..., u m ) ∈ ℝm. The system comes from understanding the stationary case of Ginzburg-Landau model. A new monotonicity formula is also set up for the following parabolic system

$$ \partial _t u_i - \Delta u_i - f_i \left( {x,u_1 , \cdots ,u_m } \right) = 0 in \left( {t_1 ,t_2 } \right) \times \mathbb{R}^n $$

, where t 1 < t 2 are two constants, (f i (x, \( \vec u \))), is given as above. The new monotonicity formulae are focused more attention on the monotonicity of nonlinear terms. The new point of the results is that an index β is introduced to measure the monotonicity of the nonlinear terms in the problems. The index β in the study of monotonicity formulae is useful in understanding the behavior of blow-up sequences of solutions. Another new feature is that the previous monotonicity formulae are extended to nonhomogeneous nonlinearities. As applications, the Ginzburg-Landau model and some different generalizations to the free boundary problems are studied.

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

References

  1. [1]

    Allard, W. K., On the first variation of a varifold, Ann. of Math., 95, 1972, 417–491.

  2. [2]

    Alt, H. W., Caffarelli, L. A. and Friedman, A., Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282, 1984, 431–462.

  3. [3]

    Bourgain, J., Brezis, H. and Mironescu, P., H ½ maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Publ. Math. Inst. Hautes Études Sci., 99, 2004, 1–115.

  4. [4]

    Caffarelli, L. A., A monotonicity formula for heat functions in disjoint domains, Boundary Value Problems for Partial Differential Equations and Applications, J. L. Lions and C. Baiocchi (eds.), Masson, Paris, 1993, 53–60.

  5. [5]

    Caffarelli, L. A., Jerison, D. and Kenig, C. E., Some new monotonicity theorems with appliacations to free boundary problems, Ann. of Math., 155, 2002, 369–402.

  6. [6]

    Caffarelli, L. A. and Kenig, C. E., Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. Math., 120, 1998, 391–439.

  7. [7]

    Caffarelli, L. A., Karp, L. and Shahgholian, H., Regularity of a free boundary with application to the Pompeiu problem, Ann. of Math., 151, 2000, 269–292.

  8. [8]

    Caffarelli, L. A. and Lin, F. H., Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21(3), 2008, 847–862.

  9. [9]

    Caffarelli, L. A., Petrosyan, A. and Shahgholian, H., Regularity of a free boundary in parabolic potential theory, J. Amer. Math. Soc., 17, 2004, 827–869.

  10. [10]

    Ecker, K., A local monotonicity formula for mean curvature flow, Ann. of Math., 154, 2001, 503–523.

  11. [11]

    Ecker, K., Local monotonicity formulas for some nonlinear diffusion equations, Calc. Var. Partial Differential Equations, 23(1), 2005, 67–81.

  12. [12]

    Fleming, W. H., On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2), 11, 1962, 69–90.

  13. [13]

    Friedman, A., Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969.

  14. [14]

    Giga, Y. and Kohn, R. V., Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math., 38, 1985, 297–319.

  15. [15]

    Hamilton, R. S., Monotonicity formulas for parabolic flows on manifolds, Comm. Anal. Geom., 1, 1993, 127–137.

  16. [16]

    Hamilton, R. S., The formation of singularities in the Ricci flow, Survey of Differential Geometry, Vol. 2, 1995, 7–136.

  17. [17]

    Huisken, G., Asymptotic behaviour for singularities of the mean curvature flow, J. Differential Geom., 31, 1990, 285–299.

  18. [18]

    Ladyzenskaja, O. A., Solonnikov V. A. and Ural’ceva, N. N., Linear and quasi-linear equations of parabolic type, Transl. Math. Monographs, Vol. 23, A. M. S., Providence, RI, 1988.

  19. [19]

    Lin, F. H., On regularity and singularity of free boundaries in obstacle problems, Chin. Ann. Math., 30B(5), 2009, 645–652.

  20. [20]

    Lin, F. H. and Riviere, T., Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc., 1, 1999, 237–311; Erratum, 2, 2000, 87–91.

  21. [21]

    Ma, L. and Su, N., Obstacle problem in scalar Ginzburg-Landau equation, J. Partial Differential Equations, 17, 2004, 49–56.

  22. [22]

    Pacard, F., Partial regularity for weak solutions of a nonlinear elliptic equation, Manuscripta Math., 79, 1993, 161–172.

  23. [23]

    Perelman, G., The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159.

  24. [24]

    Price, P., A monotonicity formula for Yang-Mills fields, Manuscripta Math., 43, 1983, 131–166.

  25. [25]

    Riviere, T., Line vortices in the U(1)-Higgs model, ESAIM Control Optim. Calc. Var., 1, 1996, 77–167.

  26. [26]

    Schoen, R. M., Analytic aspects of the harmonic map problem, Seminar on Nonlinear Partial Differential Equations, S. S. Chern (ed.), Springer-Verlag, New York, 1984, 321–358.

  27. [27]

    Simon, L. M., Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis, Vol. 3, Australian National University, Canberra, 1983.

  28. [28]

    Struwe, M., On the evolution of harmonic maps in higher dimensions, J. Differential Geom., 28, 1988, 485–502.

  29. [29]

    Weiss, G. S., Partial regularity for a minimum problem with free boundary, J. Geom. Anal., 9(2), 1999, 317–326.

  30. [30]

    Weiss, G. S., Partial regularity for weak solution of an elliptic free boundary problem, Comm. Part. Diff. Eqs., 23, 1998, 439–455.

  31. [31]

    Weiss, G. S., A homogeneity improvement approach to the obstacle problem, Invent. Math., 138, 1999, 23–50.

  32. [32]

    Weiss, G. S., Self-similar blow-up and Hausdorff dimension estimates for a class of parabolic free boundary problems, SIAM J. Math. Anal., 30, 1999, 623–644.

Download references

Author information

Correspondence to Li Ma.

Additional information

Project supported by the National Natural Science Foundation of China (No. 10631020) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20060003002).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ma, L., Song, X. & Zhao, L. New monotonicity formulae for semi-linear elliptic and parabolic systems. Chin. Ann. Math. Ser. B 31, 411–432 (2010). https://doi.org/10.1007/s11401-008-0282-8

Download citation

Keywords

  • Elliptic systems
  • Parabolic system
  • Monotonicity formula
  • Ginzburg-Landau model

2000 MR Subject Classification

  • 35Jxx
  • 17B40
  • 17B50