Analysis for non-local phase transitions close to the critical exponent $s=\frac12$

TL;DR

This paper investigates the behavior of non-local phase transition energies near the critical fractional exponent s=1/2, establishing convergence results and continuity of surface tensions as the parameter approaches the critical value.

Contribution

It provides a detailed analysis of the $eta$-convergence of fractional energies near s=1/2 and proves the continuity of the associated surface tensions across the critical exponent.

Findings

Gamma-convergence to sharp-interface functional as s approaches 1/2

Continuity of Gamma-limits with respect to s in [1/2,1)

Surface tensions are continuous functions of s

Abstract

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 $\lambda(\varepsilon,s)$, continuous in both variables, such that \[ \mathcal{F}^{s_\varepsilon}_\varepsilon(u)=\frac{\lambda(\varepsilon,s_\varepsilon)}{\varepsilon}\int W(u)dt+\lambda(\varepsilon,s_\varepsilon)\varepsilon^{(2s_\varepsilon-1)^+}[u]_{H^{s_\varepsilon}}^2 \] $\Gamma$-converge, for any choice of $s_\varepsilon \to \frac12$ as $\varepsilon\to 0$, to the sharp-interface functional found by Alberti, Bouchitt\'e 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 $\Gamma$-convergence introduced by Braides and Truskinovsky, and that the $\Gamma$-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 $[\frac12,1)$.