Higher-order non-local gradient theory of phase-transitions
Margherita Solci
TL;DR
This work analyzes a one‑dimensional phase‑transition model that augments a double‑well potential with a higher‑order fractional gradient term in $H^{k+s}$, $k\in\mathbb N$, $s\in(0,1)$, $k+s>\tfrac12$. It proves that the scaled energies $F_\varepsilon$ Gamma‑converge to a sharp‑interface functional $F(u)=m_{k+s}\#S(u)$ on $BV(I;{-1,1})$, with $m_{k+s}$ defined by a 1D optimal‑profile problem and a normalization that matches the local $H^k$ cases. The paper also addresses degeneracies at integer exponents by introducing a corrected interpolation $F^{k,s}_\varepsilon$ that yields a continuous interpolation $m_k(s)$ with $m_k$ and $m_{k+1}$ as limits when $s\to0^+$ and $s\to1^-$, respectively. The main novelty lies in unifying higher‑order local and fractional perturbations in a one‑dimensional Gamma‑convergence framework, establishing compactness, lower/upper bounds, and optimal profiles, with potential extensions to higher dimensions.
Abstract
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 $Γ$-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, Bouchitté and Seppecher, Savin and Valdinoci, and Palatucci and Vincini, in the case $k=0$ and $s\in(\frac12,1)$.