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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.17536 |
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| _version_ | 1866917505906245632 |
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| author | Geng, Zhiyuan Wang, Changyou |
| author_facet | Geng, Zhiyuan Wang, Changyou |
| contents | n this paper, we revisit the existence of global weak solutions of wave maps from $\R^n$ into the sphere $\mathbb{S}^{L-1}$, $\Box u\perp T_u \mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\R^n\times \R_+$ to $\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \cite{Giorgi1996}, which has been implemented by Serra and Tilli \cite{Serra-Tilli2012, Serra-Tilli2016} for certain class of nonlinear wave equations. This approach is also applicable to the $SO(m)$-target manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17536 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Variational Approximations For Wave Maps Geng, Zhiyuan Wang, Changyou Analysis of PDEs n this paper, we revisit the existence of global weak solutions of wave maps from $\R^n$ into the sphere $\mathbb{S}^{L-1}$, $\Box u\perp T_u \mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\R^n\times \R_+$ to $\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \cite{Giorgi1996}, which has been implemented by Serra and Tilli \cite{Serra-Tilli2012, Serra-Tilli2016} for certain class of nonlinear wave equations. This approach is also applicable to the $SO(m)$-target manifold. |
| title | On Variational Approximations For Wave Maps |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.17536 |