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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.15856 |
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| _version_ | 1866910501923979264 |
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| author | Salib, Anthony Weiss, Georg S. |
| author_facet | Salib, Anthony Weiss, Georg S. |
| contents | We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v) = \int_{B_1} \vert\nabla v\vert^2 + Per(\{v>0\},B_1).$$ In the two-phase case, we show that the energies $$E_{\varepsilon}(v) = \int_{B_1}\vert\nabla v\vert^2 + \frac{1}{\varepsilon}W\left(\frac{v}{\varepsilon^{1/2}}\right),$$ $Γ$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $[-1,1]$ . As a consequence, we show that bounded local minimisers of $E_{\varepsilon}$ converge to a local minimiser of $E$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15856 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A singular perturbation approach to the Dirichlet-area minimisation problem Salib, Anthony Weiss, Georg S. Analysis of PDEs 35R35 We study both one and two-phase minimisers of the Dirichlet-area energy $$E(v) = \int_{B_1} \vert\nabla v\vert^2 + Per(\{v>0\},B_1).$$ In the two-phase case, we show that the energies $$E_{\varepsilon}(v) = \int_{B_1}\vert\nabla v\vert^2 + \frac{1}{\varepsilon}W\left(\frac{v}{\varepsilon^{1/2}}\right),$$ $Γ$-converge to $E$ as $\varepsilon \to 0$, where $W$ is the double well potential extended by zero outside of $[-1,1]$ . As a consequence, we show that bounded local minimisers of $E_{\varepsilon}$ converge to a local minimiser of $E$. |
| title | A singular perturbation approach to the Dirichlet-area minimisation problem |
| topic | Analysis of PDEs 35R35 |
| url | https://arxiv.org/abs/2405.15856 |