Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2211.03131 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911780328964096 |
|---|---|
| author | Badran, Marco del Pino, Manuel |
| author_facet | Badran, Marco del Pino, Manuel |
| contents | We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 Δ^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$
E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$
u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^βν_β(y)).
$$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{ν_1,ν_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_03131 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds Badran, Marco del Pino, Manuel Analysis of PDEs 35J61 We consider the magnetic Ginzburg-Landau equations in a compact manifold $N$ $$ \begin{cases} -\varepsilon^2 Δ^{A} u=\frac{1}{2}(1-|u|^2)u,\\ \varepsilon^2 d^*dA=\langle\nabla^A u,iu\rangle \end{cases} $$ formally corresponding to the Euler-Lagrange equations for the energy functional $$ E(u,A)=\frac{1}{2}\int_{N}\varepsilon^2|\nabla^Au|^{2}+\varepsilon^4|dA|^{2}+\frac{1}{4}(1-|u|^{2})^{2}. $$ Here $u:N\to \mathbb{C}$ and $A$ is a 1-form on $N$. Given a codimension-2 minimal submanifold $M\subset N$ which is also oriented and non-degenerate, we construct a solution $(u_\varepsilon,A_\varepsilon)$ such that $u_\varepsilon$ has a zero set consisting of a smooth surface close to $M$. Away from $M$ we have $$ u_\varepsilon(x)\to\frac{z}{|z|},\quad A_\varepsilon(x)\to\frac{1}{|z|^2}(-z^2dz^1+z^1dz^2),\quad x=\exp_y(z^βν_β(y)). $$ as $\varepsilon\to 0$, for all sufficiently small $z\ne 0$ and $y\in M$. Here, $\{ν_1,ν_2\}$ is a normal frame for $M$ in $N$. This improves a recent result by De Philippis and Pigati who built a solution for which the concentration phenomenon holds in an energy, measure-theoretical sense. |
| title | Solutions of the Ginzburg-Landau equations concentrating on codimension-2 minimal submanifolds |
| topic | Analysis of PDEs 35J61 |
| url | https://arxiv.org/abs/2211.03131 |