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

Abstract

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