$\Gamma$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm

Abstract

We study functionals \begin{equation*} F_\varepsilon (u) := λ_\varepsilon \int_ΩW(u) \, dx + \varepsilon \|u\|_{H^{1/2}}^2 \end{equation*} for a double well potential $W$ and the Gagliardo seminorm $\Vert \cdot {\Vert }_{H^{1/2}}$ when $\epsilon \ln ({\lambda }_{\epsilon })\to k$ as $\epsilon \to 0^{+}$ and show compactness in the space of $BV$ functions on $\Omega$ and the $\Gamma$-convergence to the classical surface tension functional.

Corrected typo in Lemma 2.2, Lemma 2.4 and Corollary 2.8 (and subsequent ones)