## Abstract

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

where Ω ⊂⊂ ℝ^{n} 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* ∈ ℝ^{n} and \( \vec u \) = (*u*
_{1}, ..., *u*
_{
m
}) ∈ ℝ^{m}. 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

, 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.

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## References

- [1]
Allard, W. K., On the first variation of a varifold,

*Ann. of Math.*,**95**, 1972, 417–491. - [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]
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]
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]
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]
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]
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]
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]
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]
Ecker, K., A local monotonicity formula for mean curvature flow,

*Ann. of Math.*,**154**, 2001, 503–523. - [11]
Ecker, K., Local monotonicity formulas for some nonlinear diffusion equations,

*Calc. Var. Partial Differential Equations*,**23**(1), 2005, 67–81. - [12]
Fleming, W. H., On the oriented Plateau problem,

*Rend. Circ. Mat. Palermo*(2),**11**, 1962, 69–90. - [13]
Friedman, A., Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York, 1969.

- [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]
Hamilton, R. S., Monotonicity formulas for parabolic flows on manifolds,

*Comm. Anal. Geom.*,**1**, 1993, 127–137. - [16]
Hamilton, R. S., The formation of singularities in the Ricci flow, Survey of Differential Geometry, Vol. 2, 1995, 7–136.

- [17]
Huisken, G., Asymptotic behaviour for singularities of the mean curvature flow,

*J. Differential Geom.*,**31**, 1990, 285–299. - [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]
Lin, F. H., On regularity and singularity of free boundaries in obstacle problems,

*Chin. Ann. Math.*,**30B**(5), 2009, 645–652. - [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]
Ma, L. and Su, N., Obstacle problem in scalar Ginzburg-Landau equation,

*J. Partial Differential Equations*,**17**, 2004, 49–56. - [22]
Pacard, F., Partial regularity for weak solutions of a nonlinear elliptic equation,

*Manuscripta Math.*,**79**, 1993, 161–172. - [23]
Perelman, G., The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159.

- [24]
Price, P., A monotonicity formula for Yang-Mills fields,

*Manuscripta Math.*,**43**, 1983, 131–166. - [25]
Riviere, T., Line vortices in the

*U*(1)-Higgs model,*ESAIM Control Optim. Calc. Var.*,**1**, 1996, 77–167. - [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]
Simon, L. M., Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Analysis, Vol. 3, Australian National University, Canberra, 1983.

- [28]
Struwe, M., On the evolution of harmonic maps in higher dimensions,

*J. Differential Geom.*,**28**, 1988, 485–502. - [29]
Weiss, G. S., Partial regularity for a minimum problem with free boundary,

*J. Geom. Anal.*,**9**(2), 1999, 317–326. - [30]
Weiss, G. S., Partial regularity for weak solution of an elliptic free boundary problem,

*Comm. Part. Diff. Eqs.*,**23**, 1998, 439–455. - [31]
Weiss, G. S., A homogeneity improvement approach to the obstacle problem,

*Invent. Math.*,**138**, 1999, 23–50. - [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.

## Author information

## 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).

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### 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

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### Keywords

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

### 2000 MR Subject Classification

- 35Jxx
- 17B40
- 17B50