Energy identity for Ginzburg-Landau approximation of harmonic maps
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2025
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908290642870272 |
|---|---|
| author | Li, Xuanyu |
| author_facet | Li, Xuanyu |
| contents | Given two Riemannian manifolds $M$ and $N\subset\mathbb{R}^L$, we consider the energy concentration phenomena of the penalized energy functional $$E_ε(u)=\int_M\frac{\vert\nabla u\vert^2}{2}+\frac{F(u)}{ε^2},u\in W^{1,2}(M,\mathbb{R}^L),$$ where $F(x)$=dist$(x,N)$ in a small tubular neighborhood of $N$ and is constant away from $N$. It was shown by Chen-Struwe that as $ε\rightarrow0$, the critical points $u_ε$ of $E_ε$ with energy bound $E_ε(u_ε)\leqslantΛ$ subsequentially converge weakly in $W^{1,2}$ to a weak harmonic map $u:M\rightarrow N$ . In addition, we have the convergence of the energy density $$\left(\frac{\vert\nabla u_ε\vert^2}{2}+\frac{F(u_ε)}{ε^2}\right)dx\rightarrow\frac{\vert\nabla u_ε\vert^2}{2}dx+ν,$$ and the defect measure $ν$ above is $(dimM-2)$-rectifiable. Lin-Wang showed that if $N$ is a sphere or dim$M$=2, then the density of $ν$ can be expressed by the sum of energies of harmonic spheres. In this paper, we prove this result for an arbitrary $M$ using the idea introduced by Naber-Valtorta. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23675 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Energy identity for Ginzburg-Landau approximation of harmonic maps Li, Xuanyu Analysis of PDEs Differential Geometry 35J60, 35Q56, 58E20 Given two Riemannian manifolds $M$ and $N\subset\mathbb{R}^L$, we consider the energy concentration phenomena of the penalized energy functional $$E_ε(u)=\int_M\frac{\vert\nabla u\vert^2}{2}+\frac{F(u)}{ε^2},u\in W^{1,2}(M,\mathbb{R}^L),$$ where $F(x)$=dist$(x,N)$ in a small tubular neighborhood of $N$ and is constant away from $N$. It was shown by Chen-Struwe that as $ε\rightarrow0$, the critical points $u_ε$ of $E_ε$ with energy bound $E_ε(u_ε)\leqslantΛ$ subsequentially converge weakly in $W^{1,2}$ to a weak harmonic map $u:M\rightarrow N$ . In addition, we have the convergence of the energy density $$\left(\frac{\vert\nabla u_ε\vert^2}{2}+\frac{F(u_ε)}{ε^2}\right)dx\rightarrow\frac{\vert\nabla u_ε\vert^2}{2}dx+ν,$$ and the defect measure $ν$ above is $(dimM-2)$-rectifiable. Lin-Wang showed that if $N$ is a sphere or dim$M$=2, then the density of $ν$ can be expressed by the sum of energies of harmonic spheres. In this paper, we prove this result for an arbitrary $M$ using the idea introduced by Naber-Valtorta. |
| title | Energy identity for Ginzburg-Landau approximation of harmonic maps |
| topic | Analysis of PDEs Differential Geometry 35J60, 35Q56, 58E20 |
| url | https://arxiv.org/abs/2503.23675 |