American Institute of Mathematical Sciences

May  2013, 12(3): 1307-1319. doi: 10.3934/cpaa.2013.12.1307

The regularity for a class of singular differential equations

 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  February 2012 Revised  May 2012 Published  September 2012

We find an iteration technique and thus prove the optimal global regularity for the boundary value problem of a class of singular differential equations with strongly singular lower terms at the boundary. As applications, we obtain the regularity for the radial solutions of Ginzburg-Landau equations and harmonic maps.
Citation: Huaiyu Jian, Xiaolin Liu, Hongjie Ju. The regularity for a class of singular differential equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1307-1319. doi: 10.3934/cpaa.2013.12.1307
References:
 [1] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices," Birkhauser, 1993. doi: 10.1007/978-1-4612-0287-5. [2] K. Q. Chang, W. Y. Ding and R. G. Ye, Finite time blow-up of the heat flow of harmonic maps from spheres, J. Differential Geom., 36 (1992), 507-515. [3] T. R. Ding and C. Z. Li, "A Course on Ordinary Diferential Equations," Higher Educational Press, Beijing, 1991. [4] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Seconder Order," Springer-Verlag, Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0. [5] M. Guan, S. Gustafson and T. P. Tsai, Global existence and blow-up for harmonic map heat flow, J. Differential Equations, 246 (2009), 1-20. doi: 10.1016/j.jde.2008.09.011. [6] C. F. Gui, H. Y. Jian and H. J. Ju, Properties of translating solutions to mean curvature flow, Discrete Contin. Dyn. Syst., 28 (2010), 441-453. doi: 10.3934/dcds.2010.28.441. [7] R. M. Herve and M. Herve, Qualitative study of the real solutions of a differential equation associated with the Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 427-440. [8] H. Y. Jian, H. J. Ju and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975. [9] H. Y. Jian and Y. N. Liu, Ginzburg-Landau vortex and mean curvature flow with external force field, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1831-1842. doi: 10.1007/s10114-005-0698-y. [10] H. Y. Jian and B. Song, Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Differential Equations, 170 (2001), 123-141. doi: 10.1006/jdeq.2000.3822. [11] H. Y. Jian and X. J. Wang, Bernsterin theorem and regularity for a class of Monge Amp\ere equations, preprint, 2010. [12] H. Y. Jian and X. J. Wang, Global regularity for fully nonlinear singular elliptic equations, preprint, 2011. [13] H. Y. Jian and X. W. Xu, The vortex dynamics of a Ginzburg-Landau system under pinning effect, Science in China Ser. A, 46 (2003), 488-498. doi: 10.1007/BF02884020. [14] H. J. Ju, J. Lu and H. Y. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963. [15] P. Mironescu , On the stability of radial solutions of the Ginzburg-Landau equations, J. Funct. Anal., 130 (1995), 334-344. doi: 10.1006/jfan.1995.1073. [16] P. Raphael and R. Schweyer, Stable blow-up dynamics for 1-corotational enenrgy critical harmonic heat flow, preprint, arXiv:1106.0914

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References:
 [1] F. Bethuel, H. Brezis and F. Helein, "Ginzburg-Landau Vortices," Birkhauser, 1993. doi: 10.1007/978-1-4612-0287-5. [2] K. Q. Chang, W. Y. Ding and R. G. Ye, Finite time blow-up of the heat flow of harmonic maps from spheres, J. Differential Geom., 36 (1992), 507-515. [3] T. R. Ding and C. Z. Li, "A Course on Ordinary Diferential Equations," Higher Educational Press, Beijing, 1991. [4] D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Seconder Order," Springer-Verlag, Berlin Heidelberg, 2001. doi: 10.1007/978-3-642-61798-0. [5] M. Guan, S. Gustafson and T. P. Tsai, Global existence and blow-up for harmonic map heat flow, J. Differential Equations, 246 (2009), 1-20. doi: 10.1016/j.jde.2008.09.011. [6] C. F. Gui, H. Y. Jian and H. J. Ju, Properties of translating solutions to mean curvature flow, Discrete Contin. Dyn. Syst., 28 (2010), 441-453. doi: 10.3934/dcds.2010.28.441. [7] R. M. Herve and M. Herve, Qualitative study of the real solutions of a differential equation associated with the Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 427-440. [8] H. Y. Jian, H. J. Ju and W. Sun, Traveling fronts of curve flow with external force field, Commun. Pure Appl. Anal., 9 (2010), 975-986. doi: 10.3934/cpaa.2010.9.975. [9] H. Y. Jian and Y. N. Liu, Ginzburg-Landau vortex and mean curvature flow with external force field, Acta Math. Sin. (Engl. Ser.), 22 (2006), 1831-1842. doi: 10.1007/s10114-005-0698-y. [10] H. Y. Jian and B. Song, Vortex dynamics of Ginzburg-Landau equations in inhomogeneous superconductors, J. Differential Equations, 170 (2001), 123-141. doi: 10.1006/jdeq.2000.3822. [11] H. Y. Jian and X. J. Wang, Bernsterin theorem and regularity for a class of Monge Amp\ere equations, preprint, 2010. [12] H. Y. Jian and X. J. Wang, Global regularity for fully nonlinear singular elliptic equations, preprint, 2011. [13] H. Y. Jian and X. W. Xu, The vortex dynamics of a Ginzburg-Landau system under pinning effect, Science in China Ser. A, 46 (2003), 488-498. doi: 10.1007/BF02884020. [14] H. J. Ju, J. Lu and H. Y. Jian, Translating solutions to mean curvature flow with a forcing term in Minkowski space, Commun. Pure Appl. Anal., 9 (2010), 963-973. doi: 10.3934/cpaa.2010.9.963. [15] P. Mironescu , On the stability of radial solutions of the Ginzburg-Landau equations, J. Funct. Anal., 130 (1995), 334-344. doi: 10.1006/jfan.1995.1073. [16] P. Raphael and R. Schweyer, Stable blow-up dynamics for 1-corotational enenrgy critical harmonic heat flow, preprint, arXiv:1106.0914
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