zbMATH — the first resource for mathematics

Limiting behavior of the Ginzburg-Landau functional. (English) Zbl 1028.49015
Authors’ abstract: “We continue our study of the functional \[ \mathbb{E}_\varepsilon (u):=\int_U {1\over 2}|\nabla u|^2+{1\over 4 \varepsilon^2} \bigl(1-|u|^2 \bigr)^2dx, \] for \(u\in H^1(U; \mathbb{R}^2)\), where \(U\) is a bounded, open subset of \(\mathbb{R}^2\). Compactness results for the scaled Jacobian of \(u^\varepsilon\) are proved under the assumption that \(\mathbb{E}_\varepsilon (u^\varepsilon)\) is bounded uniformly by a function of \(\varepsilon\). In addition, the Gamma limit of \(\mathbb{E}_\varepsilon (u^\varepsilon)/ (\ln\varepsilon)^2\) is shown to be \[ \mathbb{E}(v): =\tfrac 12\|v \|^2_2+ \|\nabla \times v\|_{\mathcal M}, \] where \(v\) is the limit of \(j (u^\varepsilon)/ |\ln\varepsilon |\), \(j (u^\varepsilon): =u^\varepsilon \times Du^\varepsilon\), and \(\|\cdot \|_{\mathcal M}\) is the total variation of a Radon measure. These results are applied to the Ginzburg-Landau functional \[ \mathbb{F}_\varepsilon (u,A;h_{\text{ext}}) :=\int_U{1\over 2}|\nabla_A u|^2+{1\over 4\varepsilon^2} \bigl(1-|u|^2 \bigr)^2+ {1 \over 2}|\nabla\times A-h_{\text{ext}}|dx, \] with external magnetic field \(h_{\text{ext}} \approx H|\ln \varepsilon|\). The Gamma limit of \(\mathbb{F}_\varepsilon/ (\ln\varepsilon)^2\) is calculated to be \[ \mathbb{F}(v,a;H): = \tfrac 12\bigl [\|v-a\|^2_2+ \|\nabla\times v\|_{\mathcal M}+\|\nabla \times a-H\|^2_2 \bigr], \] where \(v\) is as before, and \(a\) is the limit of \(A^\varepsilon/|\ln\varepsilon|\)”.

49J45 Methods involving semicontinuity and convergence; relaxation
Full Text: DOI
[1] Ambrosio, L.; DeLellis, C.; Mantegazza, C., Line energies for gradient vector fields in the plane, Calc. var. partial differential equations, 9, 255-327, (1999)
[2] Bethuel, F., A characterization of maps in H1(B3;S2) which can be approximated by smooth maps, Ann. inst. H. Poincaré, 7/4, 269-286, (1990) · Zbl 0708.58004
[3] Bethuel, F.; Riviere, T., Vortices for a variational problem related to superconductivity, Ann. anal. non lineaire, 12, 243-303, (1995) · Zbl 0842.35119
[4] Bethuel, F.; Brezis, H.; Hélein, F., Ginzburg – landau vortices, (1994), Birkhauser New York · Zbl 0802.35142
[5] Brezis, H., Problémes unilatéraux, J. math. pures appl., 51, 1-168, (1972) · Zbl 0237.35001
[6] Brezis, H.; Serfaty, S., A variational formulation for the two sided obstacle problem with measure data, Comm. contemp. math., (2001)
[7] Brezis, H.; Bourgain, J.; Mironescu, P., On the structure of the Sobolev space H1/2 with values into the circle, C. R. acad. sci. Paris ser. I math., 331, 119-124, (2000) · Zbl 0970.35069
[8] Brezis J. M. Coron, H.; Lieb, E.H., Harmonic maps with defects, Comm. math. phys., 107, 649-705, (1986) · Zbl 0608.58016
[9] Chapman, S.J.; Rubinstein, J.; Schatzman, M., A Mean field model of superconducting vortices, European J. appl. math., 7, 97-111, (1996) · Zbl 0849.35135
[10] Chapman, S.J., A hierarchy of models for type-II superconductors, SIAM rev., 42, 555-598, (2000) · Zbl 0967.82014
[11] Dal Maso, G., An introduction to γ-convergence, (1993), Birkhauser Boston · Zbl 0816.49001
[12] A. DeSimone, R. Kohn, S. Muller, and, F. Otto, A compactness result in the gradient theory of phase transitions, preprint, 1999. · Zbl 0986.49009
[13] Evans, L.C.; Gariephy, R.F., Measure theory and fine properties of functions, (1992), CRC Press London
[14] Friedman, A., Variational principles and free-boundary problems, (1982), Wiley New York · Zbl 0564.49002
[15] Giaquinta, M.; Modica, G.; Soucek, J., Cartesian currents in the calculus of variations I, II, (1998), Springer-Verlag New York · Zbl 0914.49001
[16] Jerrard, R.L., Lower bounds for generalized ginzburg – landau functionals, SIAM math. anal., 30/4, 721-746, (1999) · Zbl 0928.35045
[17] Jerrard, R.L.; Soner, H.M., The Jacobian and the ginzburg – landau functional, Cal. var., 14, 151-191, (2002) · Zbl 1034.35025
[18] Jerrard, R.L.; Soner, H.M., Rectifiability of the distributional Jacobian for a class of functions, C. R. acad. sci. Paris Sér. I, 329, 683-688, (1999) · Zbl 0946.49033
[19] Jerrard, R.L.; Soner, H.M., Functions of higher bounded variation, Indiana univ. math. J., (2001) · Zbl 1057.49036
[20] Rivière, T., Lignes de tourbillons dans le modèle abelien de Higgs, C. R. acad. sci. Paris sci., 32, 73-76, (1995) · Zbl 0840.35109
[21] Rivière, T., Line vortices in the U(1)-Higgs model, Cont. opt. calc. var., 1, 77-167, (1996) · Zbl 0874.53019
[22] Rubinstein, J., Six lectures on superconductivity, (1995), American Mathematical Society Providence, p. 163-184 · Zbl 0921.35161
[23] Sandier, E., Lower bounds for the energy of unit vector fields and applications, J. funct. anal., 152/2, 379-403, (1998) · Zbl 0908.58004
[24] Sandier, E., Ginzburg – landau minimizers from \(R\)n+1 into \(R\)n and minimal connections, J. funct. anal., 152, 379-403, (1998)
[25] Sandier, E.; Serfaty, S., A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. sci. ecole norm. sup., 33, 561-592, (2000) · Zbl 1174.35552
[26] Sandier, E.; Serfaty, S., Global minimizers for the ginzburg – landau functional below the first critical magnetic field, Ann. inst. H. poincare anal. non linaire, 17, 119-145, (2000) · Zbl 0947.49004
[27] Serfaty, S., Stable configurations in superconductivity: uniqueness, multiplicity, and vortex-nucleation, Arch. rational mech. anal., 149, 329-365, (1999) · Zbl 0959.35154
[28] Schwarz, G., Hodge decomposition—A method for solving boundary value problems, (1995), Springer-Verlag Berlin · Zbl 0828.58002
[29] Tinkham, M., Introduction to superconductivity, (1996), McGraw-Hill New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.