Ginzburg-Landau equation with magnetic effect: non-simply-connected domains

Shuichi JIMBO; Jian ZHAI

doi:10.2969/jmsj/05030663

Shuichi JIMBO, Jian ZHAI

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JOURNAL FREE ACCESS

1998 Volume 50 Issue 3 Pages 663-684

DOI https://doi.org/10.2969/jmsj/05030663

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Published: 1998 Received: July 24, 1995 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Koji KUBOTA on the occasion od his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: AUTHOR Details: Wrong : Shuichi JIMBO¹⁾, Jian ZHAI²⁾
Right : Shuichi JIMBO¹⁾, Jian ZHAI¹⁾²⁾

Date of correction: October 20, 2006 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Department of Mathematics, Faculty of Science, Hokkaido University
2) Department of Mathematics, Zhejiang Unviersity

Right :
1) Department of Mathematics, Faculty of Science, Hokkaido University
2) Department of Mathematics, Zhejiang University

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. A. Adams, Sobolev Spaces, Academic Press, 1975.
2) S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727.
3) M. Berger and Y. Chen, Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal. 82 (1989), 259-295.
4) F. Bethuel, H. Brezis and F. Helein, Ginzburg Landau Vortices, Birkhäuser (1994).
5) N. A. Bobylev, Topological index of extremals of multidimensional variational problems, Funct. Anal. Appl. 20 (1986), 89-93.
6) K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Diff. Eqs. 40 (1981), 232-252.
7) A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta. Math. 88 (1952), 85-139.
8) S. Campanato, Generation of analytic semigroups in the Hölder topology by elliptic operators of second order with Neumann boundary condition, Estrato da Le Matematische 35 (1980), 61-72.
9) R. W. Carroll and A. J. Glick, On the Ginzburg-Landau equations, Arch. Rat. Mech. Anal. 16 (1968), 373-384.
10) Y. Chen, Nonsymmetric vortices for the Ginzburg-Landau equations on the bounded domain, J. Math. Phys. 30 (1989), 1942-1950.
11) Q. Du, M. Gunzberger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review 34 (1992), 54-81.
12) D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983.
13) A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhauser 1980.
14) S. Jimbo and Y. Morita, Stability of non-constant steady state solutions to a Ginzburg-Landau equation in higher space dimensions, Nonlinear Anal. TMA. 22 (1994), 753-770.
15) S. Jimbo and Y. Morita, Ginzburg-Landau equation and stable solutions in a rotational domain, SIAM J. Math. Anal. 27 (1996), 1360-1385.
16) S. Jimbo, Y. Morita and J. Zhai, Ginzburg-Landau equation and stable solutions in a nontrivial domain, Comm. PDE 20 (1995), 2093-2112.
17) V. S. Klimov, Nontrivial solutions of the Ginzburg-Landau equation, Theor. Math. Phys. 50 (1982), 383-389.
18) F. London, Superfluid, Vol. I, John Wiley and Sons, 1950.
19) T. Miyakawa, On nonstationary solutions of the Navier-Stokesequations in an exterior domain, Hiroshima Math. J. 12 (1982), 115-140.
20) S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973.
21) F. Odeh, Existence and bifurcation theorems for the Ginzburg-Landau equations, J. Math. Phys. 8 (1967), 2351-2356.
22) J. Rubinstein and P. Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, preprint.
23) Y. Yang, Existence, regularity and asymptotic behavior of the solution to the Ginzburg Landau equations on R³, Comm. Math. Phys. 123 (1989), 147-161.
24) Y. Yang, Boundary value problems of the Ginzburg-Landau equations, Proc. Roy. Soc. Edingburgh, 114A (1990), 355-365.

Right : [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
[2] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623-727.
[3] M. Berger and Y. Chen, Symmetric vortices for the Ginzburg-Landau equations of superconductivity and the nonlinear desingularization phenomenon, J. Funct. Anal. 82 (1989), 259-295.
[4] F. Bethuel, H. Brezis and F. Helein, Ginzburg Landau Vortices, Birkhäuser (1994).
[5] N. A. Bobylev, Topological index of extremals of multidimensional variational problems, Funct. Anal. Appl. 20 (1986), 89-93.
[6] K. J. Brown, P. C. Dunne and R. A. Gardner, A semilinear parabolic system arising in the theory of superconductivity, J. Diff. Eqs. 40 (1981), 232-252.
[7] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta. Math. 88 (1952), 85-139.
[8] S. Campanato, Generation of analytic semigroups in the Hölder topology by elliptic operators of second order with Neumann boundary condition, Estrato da Le Matematische 35 (1980), 61-72.
[9] R. W. Carroll and A. J. Glick, On the Ginzburg-Landau equations, Arch. Rat. Mech. Anal. 16 (1968), 373-384.
[10] Y. Chen, Nonsymmetric vortices for the Ginzburg-Landau equations on the bounded domain, J. Math. Phys. 30 (1989), 1942-1950.
[11] Q. Du, M. Gunzberger and J. Peterson, Analysis and approximation of the Ginzburg-Landau model of superconductivity, SIAM Review 34 (1992), 54-81.
[12] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, New York, 1983.
[13] A. Jaffe and C. Taubes, Vortices and Monopoles, Birkhauser 1980.
[14] S. Jimbo and Y. Morita, Stability of non-constant steady state solutions to a Ginzburg-Landau equation in higher space dimensions, Nonlinear Anal. TMA. 22 (1994), 753-770.
[15] S. Jimbo and Y. Morita, Ginzburg-Landau equation and stable solutions in a rotational domain, SIAM J. Math. Anal. 27 (1996), 1360-1385.
[16] S. Jimbo, Y. Morita and J. Zhai, Ginzburg-Landau equation and stable solutions in a nontrivial domain, Comm. PDE 20 (1995), 2093-2112.
[17] V. S. Klimov, Nontrivial solutions of the Ginzburg-Landau equation, Theor. Math. Phys. 50 (1982), 383-389.
[18] F. London, Superfluid, Vol. I, John Wiley and Sons, 1950.
[19] T. Miyakawa, On nonstationary solutions of the Navier-Stokesequations in an exterior domain, Hiroshima Math. J. 12 (1982), 115-140.
[20] S. Mizohata, The Theory of Partial Differential Equations, Cambridge Univ. Press, Cambridge, 1973.
[21] F. Odeh, Existence and bifurcation theorems for the Ginzburg-Landau equations, J. Math. Phys. 8 (1967), 2351-2356.
[22] J. Rubinstein and P. Sternberg, Homotopy classification of minimizers of the Ginzburg-Landau energy and the existence of permanent currents, preprint.
[23] Y. Yang, Existence, regularity and asymptotic behavior of the solution to the Ginzburg Landau equations on R³, Comm. Math. Phys. 123 (1989), 147-161.
[24] Y. Yang, Boundary value problems of the Ginzburg-Landau equations, Proc. Roy. Soc. Edingburgh, 114A (1990), 355-365.

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