A fractional Hadamard formula and applications

Abstract: We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $$ \lambda_{s,p}(\Omega):=\inf \left{ [u]{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(\Omega),\,\, |u|{L^p(\Omega)}=1 \right}, $$ where $s\in (0,1)$, $\Omega$ is bounded of class $C^{1,1}$ and $p\in [1, \frac{2N}{N-2s})$ if $2s<N$, $p\in [1, \infty)$ if $2s\geq N=1$. Explicitly, we derive formula for the one-sided shape derivative of the mapping $\Omega\mapsto \lambda_{s,p}(\Omega)$ under domain perturbations. In the case where $ \lambda_{s,p}(\Omega)$ admits a unique positive minimizer (e.g. $p=1$ or $p=2$), our result implies a nonlocal version of the classical variational Hadamard formula for the first eigenvalue of the Dirichlet Laplacian on $\Omega$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $\lambda_{s,p}(\Omega)$ under volume-preserving deformations, and we find that they are balls if $p\in {1}\cup [2,\infty)$. Finally, we consider the maximization problem for $\lambda_{s,p}(\Omega)$ among annular-shaped domains of fixed volume of the type $B\setminus \overline B'$, where $B$ is a fixed ball and $B'$ is ball whose position is varied within $B$. We prove that, for $p\in {1,2}$, the value $\lambda_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.