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

Tim Heilmann

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

Analysis of PDEs 2025-11-06 v2

Authors: Tim Heilmann

Abstract

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

Keywords

$\Gamma$-convergence nonlocal phase transitions $H^{1/2}$ norm Gagliardo seminorm double well potential

Cite

@article{arxiv.2507.11054,
  title  = {$\Gamma$-convergence for nonlocal phase transitions involving the $H^{1/2}$ norm},
  author = {Tim Heilmann},
  journal= {arXiv preprint arXiv:2507.11054},
  year   = {2025}
}

Comments

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

Related papers

View all related →