Analysis for non-local phase transitions close to the critical exponent $s=\frac{1}{2}$

Abstract

We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $H^s$ close to the critical exponent $s=\frac{1}{2}$. This is done by computing a scaling factor $\lambda (\epsilon ,s)$, continuous in both variables, such that \[ \mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{λ(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+λ(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2 \] $\Gamma$-converge, for any choice of $s_{\epsilon }\to \frac{1}{2}$ as $\epsilon \to 0$, to the sharp-interface functional found by Alberti, Bouchitté and Seppecher with the scaling $|\log \epsilon {|}^{-1}$. Moreover, we prove that all the values $s\in [\frac{1}{2},1)$ are regular points for the functional ${\mathcal{F}}_{\epsilon }^s$ in the sense of equivalence by $\Gamma$-convergence introduced by Braides and Truskinovsky, and that the $\Gamma$-limits as $\epsilon \to 0$ are continuous with respect to $s$. In particular, the corresponding surface tensions, given by suitable non-local optimal-profile problems, are continuous on $[\frac{1}{2},1)$.

19 pages. Ricerche mat (2025)

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