A note on the recovery sequence in the double gradient model for phase transitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909821229334528 |
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| author | Deutsch, Jakob |
| author_facet | Deutsch, Jakob |
| contents | We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica--Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form \[ E_\varepsilon(u, Ω) = \int_Ω\left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx \] for maps $ u \in H^2(Ω; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant -- namely the cell problem on the unit cube -- in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_01893 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the recovery sequence in the double gradient model for phase transitions Deutsch, Jakob Analysis of PDEs Primary: 49J45, Secondary: 35Q74, 49Q20, 74N99 We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica--Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form \[ E_\varepsilon(u, Ω) = \int_Ω\left( \frac{1}{\varepsilon} W(\nabla u) + \varepsilon |\nabla^2 u|^2 \right) dx \] for maps $ u \in H^2(Ω; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant -- namely the cell problem on the unit cube -- in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$. |
| title | A note on the recovery sequence in the double gradient model for phase transitions |
| topic | Analysis of PDEs Primary: 49J45, Secondary: 35Q74, 49Q20, 74N99 |
| url | https://arxiv.org/abs/2510.01893 |