Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HÖLDER CONVERGENCE OF THE WEAK SOLUTION TO AN EVOLUTION EQUATION OF p-GINZBURG-LANDAU TYPE

  • Lei, Yutian (INSTITUTE OF MATHEMATICS SCHOOL OF MATHEMATICS AND COMPUTER SCIENCES NANJING NORMAL UNIVERSITY)
  • Published : 2007.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

The author studies the local $H\ddot{o}lder$ convergence of the solution to an evolution equation of p-Ginzburg-Landau type, to the heat flow of the p-harmonic map, when the parameter tends to zero. The convergence is derived by establishing a uniform gradient estimation for the solution of the regularized equation.

Keywords

References

  1. F. Bethuel, H. Brezis, and F. Helein, Asymptotics for the minimization of a GinzburgLandau functional, Calc. Var. Partial Differential Equations 1 (1993), 123-138 https://doi.org/10.1007/BF01191614
  2. Y. Z. Chen and E. DiBenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math. 395 (1989), 102-131
  3. Y. M. Chen, M. C. Hong, and N. Hungerbiihler, Heat flow of p-harmonic maps with values into spheres, Math. Z. 215 (1994), no. 1, 25-35 https://doi.org/10.1007/BF02571698
  4. E. DiBenedetto and A. Friedman, Holder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1-22
  5. S. J. Ding and Z. H. Liu, Holder convergence of Ginzburg-Landau approximations to the harmonic map heat flow, Nonlinear Anal. 46 (2001), no. 6, Ser. A : Theory Methods, 807-816 https://doi.org/10.1016/S0362-546X(00)00148-6
  6. R. Jerrard and H. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal. 142 (1998), no. 2, 99-125 https://doi.org/10.1007/s002050050085
  7. Y. T. Lei, $C^{1}^{\alpha}$ convergence of a Ginzburg-Landau type minimizer in higher dimensions, Nonlinear Anal. 59 (2004), no. 4, 609-627
  8. F. H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math. 49 (1996), no. 4, 323-359 https://doi.org/10.1002/(SICI)1097-0312(199604)49:4<323::AID-CPA1>3.0.CO;2-E
  9. J. N. Zhao, Existence and nonexistance of solutions for Ut = div(l${\nabla}$ul$^{p-2}$${\nabla}$u) + f(${\nabla}$u,u,x,t), J. Math. Anal. Appl. 172 (1993), no. 1, 130-146 https://doi.org/10.1006/jmaa.1993.1012