Analysis for non-local phase transitions close to the critical exponent $s=\frac12$
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909653442494464 |
|---|---|
| author | Picerni, Marco |
| author_facet | Picerni, Marco |
| contents | We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $H^s$ close to the critical exponent $s=\frac12$. This is done by computing a scaling factor $λ(\varepsilon,s)$, continuous in both variables, such that \[ \mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{λ(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+λ(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2 \] $Γ$-converge, for any choice of $s_\varepsilon \to \frac12$ as $\varepsilon\to 0$, to the sharp-interface functional found by Alberti, Bouchitté and Seppecher with the scaling ${|\log\varepsilon|^{-1}}$. Moreover, we prove that all the values $s\in [\frac12,1 )$ are regular points for the functional $\mathcal{F}^{s}_\varepsilon$ in the sense of equivalence by $Γ$-convergence introduced by Braides and Truskinovsky, and that the $Γ$-limits as $\varepsilon\to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac12,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis for non-local phase transitions close to the critical exponent $s=\frac12$ Picerni, Marco Analysis of PDEs 35B25 (Primary) 35G20, 35R11, 49J45, 74A50, 82B26 (Secondary) We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $H^s$ close to the critical exponent $s=\frac12$. This is done by computing a scaling factor $λ(\varepsilon,s)$, continuous in both variables, such that \[ \mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{λ(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+λ(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2 \] $Γ$-converge, for any choice of $s_\varepsilon \to \frac12$ as $\varepsilon\to 0$, to the sharp-interface functional found by Alberti, Bouchitté and Seppecher with the scaling ${|\log\varepsilon|^{-1}}$. Moreover, we prove that all the values $s\in [\frac12,1 )$ are regular points for the functional $\mathcal{F}^{s}_\varepsilon$ in the sense of equivalence by $Γ$-convergence introduced by Braides and Truskinovsky, and that the $Γ$-limits as $\varepsilon\to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac12,1)$. |
| title | Analysis for non-local phase transitions close to the critical exponent $s=\frac12$ |
| topic | Analysis of PDEs 35B25 (Primary) 35G20, 35R11, 49J45, 74A50, 82B26 (Secondary) |
| url | https://arxiv.org/abs/2502.04145 |