On nonlocal minimal graphs (Q2048703)

The authors deal with a functional analytic setting for studying nonlocal minimal graphs. This is done by introducing a (fractional, nonlocal) \(F_s\) version of the area functional defined (in an open set \(\Omega\subseteq\mathbb{R}^n\) and given \(s\in (0,1)\)) by: \[ F_s(u,\Omega):=\iint_{Q(\Omega)} G_s\left(\frac{u(x)-u(y)}{|x-y|}\right)\frac{dx\,dy}{|x-y|^{n-1+s}}, \] with \(Q(\Omega):=(\mathbb{R}^n)^2\setminus (\complement\Omega)^2\) and where \(G_s\) is defined by: \[G_s(t):=\int_0^t d\tau\int_0^\tau \frac{d\sigma}{(1+\sigma^2)^{\frac{n+1+s}{2}}}.\] The paper then provides a study of the functional \(F_s\) and its minimizers by addressing e.g. the relationship between \(F_s\) and the fractional \(s\)-perimeter of \textit{L. Caffarelli} et al. [Commun. Pure Appl. Math. 63, No. 9, 1111--1144 (2010; Zbl 1248.53009)], existence and uniqueness of minimizers (obtained under mild assumptions on the exterior data), and a priori estimates, weak and viscosity solutions of the Euler-Lagrange equation associated with \(F_s\).