Higher-order non-local gradient theory of phase-transitions

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 $$\frac{1}{\varepsilon}\int_I W(u(x))dx+\varepsilon^{2(k+s)-1}\frac{s(1-s)}{2^{1-s}}\int_{I\times I} \frac{|u^{(k)}(x)-u^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy$$ defined on the higher-order fractional Sobolev space $H^{k+s}(I)$, where $W$ is a double-well potential, $k\in \mathbb N$ and $s\in(0,1)$ with $k+s>\frac12$. We show that these functionals $\Gamma$-converge as $\varepsilon\to 0$ 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{equation*} m_{k+s} =\inf\Big\{\int_{\mathbb R} W(v)dx+\frac{s(1-s)}{2^{1-s}}\int_{\mathbb R^2}\frac{|v^{(k)}(x)-v^{(k)}(y)|^2}{|x-y|^{1+2s}} dx\,dy : v\in H^{k+s}_{\rm loc}(\mathbb R), \lim_{x\to\pm\infty}v(x)=\pm1\Big\}. \end{equation*} The normalization coefficient $\frac{s(1-s)}{2^{1-s}}$ is such that $m_{k+s}$ interpolates continuously the corresponding $m_k$ defined on standard higher-order Sobolev space $H^k(I)$, obtained by Modica and Mortola in the case $k=1$, Fonseca and Mantegazza in the case $k=2$ and Brusca, Donati and Solci for $k\ge 3$. The results also extends previous works by Alberti, Bouchitt\'e and Seppecher, Savin and Valdinoci, and Palatucci and Vincini, in the case $k=0$ and $s\in(\frac12,1)$.