We study the asymptotic behaviour of double-well energies perturbed by a higher-order fractional term, which, in the one-dimensional case, take the form \(\begin{aligned} \frac{1}{\varepsilon }\int _I W(u(x))dx+\varepsilon ^{2(k+s)-1}c_s\int _{I\times I} \frac{|u^{(k)}(x)-u^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy, \end{aligned}\) where \(u^{(k)}\) denotes the weak k-th derivative of the function u, defined on the higher-order fractional Sobolev space \(H^{k+s}(I)\) . The function W is a double-well potential with minima in \(-1\) and 1, \(k\in {\mathbb {N}}\) , \(s\in (0,1)\) with \(k+s>\frac{1}{2}\) and \(c_s>0\) . We show that these functionals \(\Gamma \) -converge to a sharp-interface functional with domain \(BV(I;\{-1,1\})\) of the form \(m_{k+s}\#(S(u))\) , with \(m_{k+s}\) given by the optimal-profile problem \(\begin{aligned} & m_{k+s} =\inf \Big \{\int _{{\mathbb {R}}} W(v)dx+c_s\int _{{\mathbb {R}}^2}\frac{|v^{(k)}(x)-v^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy:\\ & \hspace{142.26378pt}v\in H^{k+s}_\textrm{loc}({\mathbb {R}}), \lim _{x\rightarrow \pm \infty }v(x)=\pm 1\Big \}. \end{aligned}\) If we choose the coefficient \(c_s=\frac{s(1-s)}{2^{1-s}}\) , then \(m_{k+s}\) interpolates continuously the corresponding \(m_k\) defined on standard higher-order Sobolev space \(H^k(I)\) , obtained by Modica–Mortola in the case \(k=1\) (Boll Un Mat Ital B (5), 14(1):285–299, 1977), Fonseca–Mantegazza in the case \(k=2\) (SIAM J Math Anal 31(5):1121–1143, 2000) and Brusca et al. for \(k\ge 3\) (SIAM J Math Anal 57(3): 3146-3170, 2025). The results also extends previous works by Alberti et al. (C R Acad Sci Paris Sér I Math 319(4):333–338, 1994), Savin–Valdinoci (Ann Inst H Poincaré C Anal Non Linéaire 29(4), 479–500, 2012) and Palatucci–Vincini (Matematiche (Catania) 75(1):195–220, 2020) in the case \(k=0\) and \(s\in (\frac{1}{2},1)\) .
