An $L^{\mathrm{\infty }}$-variational problem involving the Fractional Laplacian

For $s\in (0,1)$ and an open bounded set $\mathrm{\Omega }\subset {\mathbb{R}}^n$, we prove existence and uniqueness of absolute minimisers of the supremal functional $$E_{\mathrm{\infty }}(u)=\mathrm{\parallel }(-\mathrm{\Delta }{)}^{s}u{\mathrm{\parallel }}_{L^{\mathrm{\infty }}({\mathbb{R}}^n)},$$ where $(-\mathrm{\Delta }{)}^s$ is the Fractional Laplacian of order $s$ and $u$ has prescribed Dirichlet data in the complement of $\mathrm{\Omega }$. We further show that the minimiser $u_{\mathrm{\infty }}$ satisfies the (fractional) PDE $$(-\mathrm{\Delta }{)}^s{u}_{\mathrm{\infty }}=E_{\mathrm{\infty }}(u_{\mathrm{\infty }})\mathrm{s}\mathrm{g}\mathrm{n}{f}_{\mathrm{\infty }}\text{qquadin}\mathrm{\Omega },$$ for some analytic function $f_{\mathrm{\infty }}\in L^1(\mathrm{\Omega })$ obtained as the restriction of an $s$-harmonic measure $\mu$ in $\mathrm{\Omega }$.