Density estimates for a variational model driven by the Gagliardo norm

Abstract

We prove density estimates for level sets of minimizers of the energy $$\eps^{2s}\|u\|_{H^s(Ω)}^2+\int_ΩW(u)\,dx,$$ with $s\in (0,1)$, where $\Vert u{\Vert }_{H^s(\Omega )}$ denotes the total contribution from $\Omega$ in the $H^s$ norm of $u$, and $W$ is a double-well potential. As a consequence we obtain, as $\eps \to 0$, the uniform convergence of the level sets of $u$ to either a $H^s$-nonlocal minimal surface if $s\in (0,\frac{1}{2})$, or to a classical minimal surface if $s\in [\frac{1}{2},1)$.

References in corpus (1)

• Gamma-convergence of nonlocal perimeter functionals

Cited by in corpus (1)

• Density estimates for phase transitions with a trace