## A boundary value problem related to the Ginzburg-Landau model.(English)Zbl 0742.35045

Summary: We analyze the Ginzburg-Landau equation for a superconductor in the case of a 2-dimensional model: a cylindrical conductor with a magnetic field parallel to the axis. This amounts to find the extrema of the free energy ${\mathcal A}_ \kappa=1/2\int_ \Omega[|(\nabla-iA)\Phi|^ 2+| B_ A|^ 2+\kappa/4(|\Phi|^ 2-1)^ 2]dx,$ where $$\Omega$$ is a bounded domain with smooth boundary in $$\mathbb{R}^ 2$$, $$A=(A_ 1,A_ 2)$$ the vector potential, $$B_ A=\partial_ 1A_ 2- \partial_ 2A_ 1$$ the magnetic field, $$\Phi$$ a complex field. We describe the connected components of the maximal configuration space, i.e. of the set of all $$(A,\Phi)$$ with components in the Sobolev space $$H^ 1(\Omega)$$ and such that $$|\Phi|=1$$ on the boundary, modulo the action of the gauge group. In the critical case $$\kappa=1$$ we give a complete description of the minimal configurations in each component.

### MSC:

 35Q40 PDEs in connection with quantum mechanics
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### References:

 [1] Berger, M.: Nonlinearity and functional analysis. New York: Academic Press 1977 · Zbl 0368.47001 [2] Bogomol’nyi, E. B.: J. Iardernoi Fiz.24, 449 (1976) [3] Deny, J., Lions, J.-L.: Ann. Inst. FourierV, 305–370 (1953) [4] Georgescu, V.: Ann. Mat. Pura. Appl.122, 159–198 (1979) · Zbl 0432.58026 [5] Georgescu, V.: Arch. Rat. Mech. Anal.74, 143–164 (1980) · Zbl 0457.53020 [6] Ghinzburg, V. I., Landau, L. D.: J. Exp. i Teoret. Fiz.20, (12), (1950) [7] Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0361.35003 [8] Jaffe, A., Taubes, C.: Vortices and monopoles: Structure of static gauge theories, Boston: Birkhaüser 1980 · Zbl 0457.53034 [9] Lions, J.-L., Magenes, E.: Ann. Sc. Norm. Sup. Pisca16, 1–44 (1962) [10] Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications, I and II. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0227.35001 [11] Morrey, Ch.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966 · Zbl 0142.38701 [12] Rose-Innes, A. C., Rhoderich, E. N.: Introduction to superconductivity. London: Pergamon Press 1978 [13] Saint-James, D., Sarma, G., Thomas, E. J.: Type II superconductivity. London: Pergamon Press 1969 [14] Schrieffer, J. R.: Theory of superconductivity New York, Amsterdam: Benjamin 1964 · Zbl 0125.24102 [15] Stein, E.: Singular Integrals and differentiability Properties of functions. Princeton: Princeton University Press, 1970 · Zbl 0207.13501 [16] Boutet de Monvel, A., Georgescu, V., Purice, R.: Sur un problème aux limites de la théorie de Ginzburg-Landau. C. R. Acad. Sci. Paris307, série I, 55–58 (1988) · Zbl 0696.35058
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