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\(H^{1/2}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation. (English) Zbl 1051.49030

Let \(G\) be a smooth bounded domain in \({\mathbb R}^3\) with \(\Omega=\partial G\) simply connected. This paper deals with the study of the manifold \(H^{1/2}(\Omega ;S^1)=\{g\in H^{1/2}(\Omega ;{\mathbb R}^2)\); \(| g| =1\) a.e. on \(\Omega\}\). Using a density result of T. Rivière, [Ann. Global Anal. Geom. 18, No. 5, 517–528 (2000; Zbl 0960.35022)], the authors prove that for any \(g\in H^{1/2}(\Omega;S^1)\), there exist two sequences \((P_i)\) and \((N_i)\) in \(\Omega\) such that \(\sum_i| P_i-N_i| <\infty\) and \(\langle T(g),\varphi\rangle =2\pi\sum_i(\varphi (P_i)-\varphi(N_i))\) for all \(\varphi\in Lip\,(\Omega;{\mathbb R})\), where \(T(g)\in{\mathcal D}'(\Omega;{\mathbb R})\) is the distribution associated to \(g\in H^{1/2}(\Omega;S^1)\) [and which describes the location and the topological degree of the singularities of \(g\)]. Moreover, if the distribution \(T(g)\) is a measure (of finite total mass), then \(T(g)=2\pi\sum_{\text{finite}}d_j\delta_{a_j}\), with \(d_j\in{\mathbb Z}\) and \(a_j\in\Omega\). Next, the authors establish a representation formula for functions belonging to \(Y:=\overline{C^\infty(\Omega;S^1)}^{H^{1/2}}\). More precisely, it is shown that for every \(g\in Y\) there exists \(\varphi\in H^{1/2}(\Omega;{\mathbb R})+W^{1,1}(\Omega;{\mathbb R})\), which is unique (modulo \(2\pi\)), such that \(g=e^{i\varphi}\). Conversely, if \(g\in H^{1/2}(\Omega;S^1)\) can be written as \(g=e^{i\varphi}\) with \(\varphi\in H^{1/2}(\Omega;{\mathbb R})+W^{1,1}(\Omega;{\mathbb R})\), then \(g\in Y\). The proof of this result relies on the notion of paraproduct, in the sense of J.-M. Bony and Y. Meyer. The heart of the matter is the estimate \[ \| \varphi\| _{H^{1/2}+W^{1,1}}\leq C_\Omega\| e^{i\varphi}\| _{H^{1/2}}(1+\| e^{i\varphi}\| _{H^{1/2}}), \] which holds for any smooth real-valued function \(\varphi\). The same estimate enables the authors to prove that for all \(g\in H^{1/2}(\Omega;S^1)\) there exists \(\varphi\in H^{1/2}(\Omega;{\mathbb R})+BV(\Omega;{\mathbb R})\) such that \(g=e^{i\varphi}\), where \(\varphi\) is not unique. Several interesting links between all possible liftings of \(g\) and the minimal connection of \(g\) are also established in the paper.
In the second part of the present work it is studied the Ginzburg-Landau energy functional \(E_\varepsilon (u)=2^{-1}\int_G| \nabla u| ^2+(4\varepsilon^2)^{-1}\int_G(| u| ^2-1)^2\), where \(\varepsilon>0\) is a small parameter. Denote \(e_{\varepsilon ,g}=\min_{H^1_g(G;{\mathbb R}^2)}E_\varepsilon (u)\) and let \(u_\varepsilon\) be an arbitrary solution of this minimization problem. Some of the main results established by the authors are the following: (i) \(e_{\varepsilon ,g}=\pi L_G(g)\log (1/\varepsilon)+o(\log (1/\varepsilon))\) as \(\varepsilon\rightarrow 0\), where \(L(g)=(2\pi)^{-1}\sup\{\langle T(g),\varphi\rangle;\;\varphi\in Lip\, (\Omega;{\mathbb R}),\;| \varphi| _{Lip}\leq 1\}\); (ii) for any \(g\in H^{1/2}(\Omega ;S^1)\), \(\| u_\varepsilon\| _{W^{1,p}(G)}\leq C_p\), for all \(1\leq p<3/2\); (iii) for every \(g\in H^{1/2}(\Omega ;S^1)\), the family \((u_\varepsilon)\) is relatively compact in \(W^{1,p}\) (\(p<3/2\)); (iv) for every \(g\in Y\), \(u_\varepsilon\rightarrow u_*=e^{i\overline\varphi}\) in \(W^{1,p}(G)\cap C^\infty (G)\), for all \(p<3/2\), where \(\overline\varphi\) is the harmonic extension of \(\varphi\).
The paper is excellently written and the results offer a much better understanding of the Ginzburg-Landau equation in dimension 3. The proofs combine powerful tools in variational calculus and modern nonlinear analysis and they rely on refined elliptic estimates, paraproducts, topological degree properties, asymptotic analysis, etc. Some interesting open problems are also raised in the paper. In the reviewer’s opinion, the paper opens several fundamental research directions in the Ginzburg-Landau theory, with applications in supraconductivity, superfluids, and minimal connections.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
35B40 Asymptotic behavior of solutions to PDEs
35J60 Nonlinear elliptic equations
46T05 Infinite-dimensional manifolds
47H11 Degree theory for nonlinear operators
49J35 Existence of solutions for minimax problems
82D50 Statistical mechanics of superfluids
82D55 Statistical mechanics of superconductors
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0960.35022
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References:

[1] R. A. Adams, Sobolev spaces, Acad. Press, 1975. · Zbl 0314.46030
[2] F. Almgren, W. Browder, and E. H. Lieb, Co-area, liquid crystals and minimal surfaces, in: Partial differential equations (Tianjin, 1986), Lect. Notes Math. 1306, Springer, 1988. · Zbl 0645.58015
[3] F. Bethuel, A characterization of maps in H1(B3,S2) which can be approximated by smooth maps, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 7 (1990), 269–286.
[4] F. Bethuel, Approximations in trace spaces defined between manifolds, Nonlinear Anal. Theory Methods Appl., 24 (1995), 121–130. · Zbl 0824.58011
[5] F. Bethuel, J. Bourgain, H. Brezis, and G. Orlandi, W1,p estimate for solutions to the Ginzburg–Landau equation with boundary data in H1/2, C. R. Acad. Sci., Paris, Sér. I, Math., 333 (2001), 1069–1076. · Zbl 1080.35020
[6] F. Bethuel, H. Brezis, and J.-M. Coron, Relaxed energies for harmonic maps, in: H. Berestycki, J.-M. Coron, and I. Ekeland (eds.), Variational Problems, pp. 37–52, Birkhäuser, 1990. · Zbl 0793.58011
[7] F. Bethuel, H. Brezis, and F. Hélein, Asymptotics for the minimization of a Ginzburg–Landau functional, Calc. Var. Partial Differ. Equ., 1 (1993), 123–148. · Zbl 0834.35014
[8] F. Bethuel, H. Brezis, and G. Orlandi, Small energy solutions to the Ginzburg–Landau equation, C. R. Acad. Sci., Paris, Sér. I, 331 (2000), 763–770. · Zbl 0969.35055
[9] F. Bethuel, H. Brezis, and G. Orlandi, Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal., 186 (2001), 432–520. · Zbl 1077.35047
[10] F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Anal., 80 (1988), 60–75. · Zbl 0657.46027
[11] J. Bourgain and H. Brezis, On the equation div Y=f and application to control of phases, J. Am. Math. Soc., 16 (2003), 393–426. · Zbl 1075.35006
[12] J. Bourgain, H. Brezis, and P. Mironescu, Lifting in Sobolev spaces, J. Anal. Math., 80 (2000), 37–86. · Zbl 0967.46026
[13] J. Bourgain, H. Brezis, and P. Mironescu, On the structure of the Sobolev space H1/2 with values into the circle, C. R. Acad. Sci., Paris, Sér. I, 310 (2000), 119–124.
[14] J. Bourgain, H. Brezis, and P. Mironescu, Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman, and A. Sulem (eds.), Optimal Control and Partial Differential Equations, pp. 439–455, IOS Press, 2001. · Zbl 1103.46310
[15] J. Bourgain, H. Brezis, and P. Mironescu, Limiting embedding theorems for Ws,p when \(s\nearrow1\) and applications, J. Anal. Math., 87 (2002), 77–101. · Zbl 1029.46030
[16] J. Bourgain, H. Brezis, and P. Mironescu, Lifting, degree and distibutional Jacobian revisited, to appear in Commun. Pure Appl. Math. · Zbl 1077.46023
[17] A. Boutet de Monvel, V. Georgescu, and R. Purice, A boundary value problem related to the Ginzburg–Landau model, Commun. Math. Phys., 142 (1991), 1–23. · Zbl 0742.35045
[18] H. Brezis, Liquid crystals and energy estimates for S2-valued maps, in: J. Ericksen and D. Kinderlehrer (eds.), Theory and Applications of Liquid Crystals, pp. 31–52, Springer, 1987.
[19] H. Brezis, J.-M. Coron, and E. Lieb, Harmonic maps with defects, Commun. Math. Phys., 107 (1986), 649–705. · Zbl 0608.58016
[20] H. Brezis, Y. Y. Li, P. Mironescu, and L. Nirenberg, Degree and Sobolev spaces, Topol. Methods Nonlinear Anal., 13 (1999), 181–190. · Zbl 0956.46024
[21] H. Brezis and P. Mironescu, Gagliardo-Nirenberg, composition and products in fractional Sobolev spaces, J. Evolution Equ., 1 (2001), 387–404. · Zbl 1023.46031
[22] H. Brezis and L. Nirenberg, Degree Theory and BMO, Part I: Compact manifolds without boundaries, Sel. Math., 1 (1995), 197–263. · Zbl 0852.58010
[23] A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, Harmonic analysis of the space BV, Rev. Mat. Iberoam.,19 (2003), 235–263. · Zbl 1044.42028
[24] F. Demengel, Une caractérisation des fonctions de W1,1(Bn,S1) qui peuvent être approchées par des fonctions régulières, C. R. Acad. Sci., Paris, Sér. I, 310 (1990), 553–557.
[25] M. Escobedo, Some remarks on the density of regular mappings in Sobolev classes of SM-valued functions, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 127–144. · Zbl 0678.46028
[26] H. Federer, Geometric measure theory, Springer, 1969. · Zbl 0176.00801
[27] M. Giaquinta, G. Modica, and J. Soucek, Cartesian Currents in the Calculus of Variations, vol. II, Springer, 1998. · Zbl 0914.49001
[28] F. B. Hang and F. H. Lin, A remark on the Jacobians, Comm. Contemp. Math.,2 (2000), 35–46. · Zbl 1033.49047
[29] R. Hardt, D. Kinderlehrer, and F. H. Lin, Stable defects of minimizers of constrained variational principles, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 5 (1988), 297–322. · Zbl 0657.49018
[30] T. Iwaniec, C. Scott, and B. Stroffolini, Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl., 157 (1999), 37–115. · Zbl 0963.58003
[31] R. L. Jerrard and H. M. Soner, Rectifiability of the distributional Jacobian for a class of functions, C. R. Acad. Sci., Paris, Sér. I, 329 (1999), 683–688. · Zbl 0946.49033
[32] R. L. Jerrard and H. M. Soner, Functions of bounded higher variation, Indiana Univ. Math. J., 51 (2002), 645–677. · Zbl 1057.49036
[33] R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differ. Equ., 14 (2002), 151–191. · Zbl 1034.35025
[34] F. H. Lin and T. Rivière, Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc., 1 (1999), 237–311; Erratum2 (2002), 87–91. · Zbl 0939.35056
[35] V. Maz’ya and T. Shaposhnikova, On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230–238. · Zbl 1028.46050
[36] A. Ponce, On the distributions of the form \(\sum_{i}(\delta_{p_{i}} - \delta_{n_{i}})\), J. Funct. Anal.,210 (2004), 391–435; part of the results were announced in a note by the same author: On the distributions of the form \(\sum_{i}(\delta_{p_{i}} - \delta_{n_{i}})\), C. R. Acad. Sci., Paris Sér. I, Math., 336 (2003), 571–576.
[37] T. Rivière, Line vortices in the U(1)-Higgs model, Control Optim. Calc. Var., 1 (1996), 77–167.
[38] T. Rivière, Dense subsets of H1/2(S2;S1), Ann. Global Anal. Geom., 18 (2000), 517–528.
[39] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal., 152 (1998), 379–403. · Zbl 0908.58004
[40] E. Sandier, Ginzburg–Landau minimizers from Rn+1 to Rn and minimal connections, Indiana Univ. Math. J., 50 (2001), 1807–1844. · Zbl 1034.58016
[41] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J. Differ. Geom., 18 (1983), 253–268. · Zbl 0547.58020
[42] L. Simon, Lectures on geometric measure theory, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. · Zbl 0546.49019
[43] D. Smets, On some infinite sums of integer valued Dirac’s masses, C. R. Acad. Sci., Paris, Sér. I, 334 (2002), 371–374. · Zbl 1154.46308
[44] V. A. Solonnikov, Inequalities for functions of the classes \({\vec W}_{p} (\mathbf{R}^{\mathbf{n}})\), J. Soviet Math., 3 (1975), 549–564. · Zbl 0349.46037
[45] H. Triebel, Interpolation theory. Function spaces. Differential operators, Johann Ambrosius Barth, Heidelberg, Leipzig, 1995. · Zbl 0830.46028
[46] G. Alberti, S. Baldo, and G. Orlandi, Variational convergence for functionals of Ginzburg–Landau type, to appear. · Zbl 1160.35013
[47] F. Bethuel, G. Orlandi, and D. Smets, On an open problem for Jacobians raised by Bourgain, Brezis and Mironescu, C. R. Acad Sci., Paris, Sér. I, 337 (2003), 381–385. · Zbl 1113.35315
[48] F. Bethuel, G. Orlandi, and D. Smets, Approximation with vorticity bounds for the Ginzburg–Landau functional, to appear in Comm. Contemp. Math. · Zbl 1129.35329
[49] J. Bourgain and H. Brezis, New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Acad Sci., Paris, Sér. I, 338 (2004), 539–543. · Zbl 1101.35013
[50] H. Federer and W. H. Fleming, Normal and integral currents, Ann. Math., 72 (1960), 458–520. · Zbl 0187.31301
[51] A. Ponce, An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., 6 (2004), 1–15. · Zbl 1051.46019
[52] J. Van Schaftingen, On an inequality of Bourgain, Brezis and Mironescu, C. R. Acad Sci., Paris, Sér. I, 338 (2004), 23–26. · Zbl 1188.26015
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