\( \Gamma \)-convergence of the Ginzburg-Landau functional with tangential boundary conditions (Q6611758)

The authors consider a bounded, connected and open set \(\Omega \subset \mathbb{R}^{2}\) with a \(C^{2,1}\)-boundary \(\partial \Omega \) having \(b+1\) connected components, with \(b\geq 0\), and the Ginzburg-Landau energy defined as: \(E_{\varepsilon }(u,\Omega )=\int_{\Omega }(\frac{1}{2}\left\vert \nabla u\right\vert ^{2}+\frac{1}{4}\varepsilon ^{2}(\left\vert u\right\vert ^{2}-1)^{2})\), on the spaces \(W_{T}^{1,2}(\Omega ;\mathbb{R} ^{2})=\{W^{1,2}(\Omega ;\mathbb{R}^{2})=u_{T}=u\cdot \tau \}\) and \( W_{N}^{1,2}(\Omega ;\mathbb{R}^{2})=\{W^{1,2}(\Omega ;\mathbb{R} ^{2})=u_{N}=u\cdot \nu \}\). \N\NThe main result of the paper proves that if a sequence \(\{u_{\varepsilon }\}_{\varepsilon \in (0,1]}\subset W_{T}^{1,2}(\Omega ;\mathbb{R}^{2})\) satisfies \(E_{\varepsilon }(u_{\varepsilon })\leq C|\log(\varepsilon )|\) for all \(\varepsilon \in (0,1]\) and some \(C>0\), then, up to a subsequence, there exists a signed Radon measure \(J_{\ast }\), supported on \(\Omega \) such that \(\lim_{\varepsilon \rightarrow 0}\left\Vert \star J(u_{\varepsilon })-J_{\ast }\right\Vert _{(C^{0,\alpha }(\Omega ))^{\ast }}=0\), for all \(0<\alpha \leq 1\), where \( \star J\) is defined through: \(\left\langle \star J(u),\varphi \right\rangle =\int_{\Omega }\varphi J(u)\), \(\varphi \in C^{0,\alpha }(\Omega )\) and \( Ju=\det(\nabla u)\). The authors characterize \(J_{\ast }\) in terms of interior defects and of boundary defects around the connected components of \( \partial \Omega \). \N\NFor the proof, the authors start recalling geometric notions concerning the coordinates, an interpolation result and a slicing result. They use the reflection technique presented by \textit{R. Jerrard} et al. [Commun. Math. Phys. 249, No. 3, 549--577 (2004; Zbl 1065.58012)]. They show that the compactness result proved by \textit{R. L. Jerrard} and \textit{H. M. Soner} [Calc. Var. Partial Differ. Equ. 14, No. 2, 151--191 (2002; Zbl 1034.35025)] is still valid in the present case.\ They finally prove a lower bound result, then an upper bound result using zeroth order \(\Gamma \)-convergence tools for the canonical harmonic map with prescribed singularities which has normal part zero.