On the Convergence of Solutions for the Ginzburg-Landau Equation and System
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916946359877632 |
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| author | Hadiji, Rejeb Han, Jongmin |
| author_facet | Hadiji, Rejeb Han, Jongmin |
| contents | Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial Ω$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial Ω$. We show that, if there exists a constant $C_1 > 0$ such that for $\varepsilon$ sufficiently small we have $\frac{1}{2} \int_Ω|\nabla u_\ve|^2 dx \leq C_1 \leq \frac{1}{2} \int_Ω|\nabla u_0|^2 dx,$ then $C_1 = \frac{1}{2} \int_Ω|\nabla u_0|^2 dx$ and
$u_\ve ~\to ~ u_0 \qin H^1(\Om)$.
We also prove that if there is a constant $C_2$ such that for $\ve$ small enough we have $ \frac12 \int_\Om |\nabla u_\ve|^2 dx \geq C_2 > \frac12 \int_\Om |\nabla u_0|^2 dx,$ then $|u_{\ve}|$ does not converge uniformly to $1$ on $\overline{\Om} $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09231 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Convergence of Solutions for the Ginzburg-Landau Equation and System Hadiji, Rejeb Han, Jongmin Analysis of PDEs 35B40, 35J60, 35Q60 Let $(u_\varepsilon)$ be a family of solutions of the Ginzburg--Landau equation with boundary condition $u_\varepsilon = g$ on $\partial Ω$ and of degree $0$. Let $u_0$ denote the harmonic map satisfying $u_0 = g$ on $\partial Ω$. We show that, if there exists a constant $C_1 > 0$ such that for $\varepsilon$ sufficiently small we have $\frac{1}{2} \int_Ω|\nabla u_\ve|^2 dx \leq C_1 \leq \frac{1}{2} \int_Ω|\nabla u_0|^2 dx,$ then $C_1 = \frac{1}{2} \int_Ω|\nabla u_0|^2 dx$ and $u_\ve ~\to ~ u_0 \qin H^1(\Om)$. We also prove that if there is a constant $C_2$ such that for $\ve$ small enough we have $ \frac12 \int_\Om |\nabla u_\ve|^2 dx \geq C_2 > \frac12 \int_\Om |\nabla u_0|^2 dx,$ then $|u_{\ve}|$ does not converge uniformly to $1$ on $\overline{\Om} $. We obtain analogous results for both symmetric and non-symmetric two-component Ginzburg--Landau systems. |
| title | On the Convergence of Solutions for the Ginzburg-Landau Equation and System |
| topic | Analysis of PDEs 35B40, 35J60, 35Q60 |
| url | https://arxiv.org/abs/2509.09231 |