On the triple junction problem on the plane without symmetry hypotheses
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917619417743360 |
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| author | Alikakos, Nicholas D. Geng, Zhiyuan |
| author_facet | Alikakos, Nicholas D. Geng, Zhiyuan |
| contents | We investigate the Allen-Cahn system
\begin{equation*}
Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2,
\end{equation*}
where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem
\begin{equation*}
\min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1.
\end{equation*}
The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_17134 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the triple junction problem on the plane without symmetry hypotheses Alikakos, Nicholas D. Geng, Zhiyuan Analysis of PDEs 35J47, 35R35, 35J50, 35B08 We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution. |
| title | On the triple junction problem on the plane without symmetry hypotheses |
| topic | Analysis of PDEs 35J47, 35R35, 35J50, 35B08 |
| url | https://arxiv.org/abs/2210.17134 |