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.