Analysis for non-local phase transitions close to the critical exponent $$s=\frac{1}{2}$$
Marco Picerni · Ricerche di Matematica · 2025
Abstract We analyze the behaviour of double-well energies perturbed by fractional Gagliardo squared seminorms in $$H^s$$ H s close to the critical exponent $$s=\frac{1}{2}$$ s = 1 2 . This is done by computing a scaling factor $$\lambda (\varepsilon ,s)$$ λ ( ε , s ) , continuous in both variables, such that $$\begin{aligned} {\mathcal {F}}^{s_\varepsilon }_\varepsilon (u)=\frac{\lambda (\varepsilon ,s_\varepsilon )}{\varepsilon }\int W(u)\,dt+\lambda (\varepsilon ,s_\varepsilon )\varepsilon ^{(2s_\varepsilon -1)^+} {[}u ]_{{s_\varepsilon }}^2 \end{aligned}$$ F ε s ε ( u ) = λ ( ε , s ε ) ε ∫ W ( u ) d t + λ ( ε , s ε ) ε ( 2 s ε - 1 ) + [ u ] s ε 2 $$\Gamma $$ Γ -converge, for any choice of $$s_\varepsilon \rightarrow \frac{1}{2}$$ s ε → 1 2 as $$\varepsilon \rightarrow 0$$ ε → 0 , to the sharp-interface functional found by Alberti, Bouchitté and Seppecher in [1] with the scaling $${|\log \varepsilon |^{-1}}$$ | log ε | - 1 . Moreover, we prove that all the values $$s\in [\frac{1}{2},1 )$$ s ∈ [ 1 2 , 1 ) are regular points for the functional $${\mathcal {F}}^{s}_\varepsilon $$