A fractional Hadamard formula and applications

We consider the domain dependence of the best constant in the subcritical fractional Sobolev constant, $λ_{s,p}(Ω):=\inf \left\{ [u]_{H^s(\mathbb{R}^N)}^2,\,\, u\in C^\infty_c(Ω),\,\, \|u\|_{L^p(Ω)}=1 \right\},$ where $s\in (0,1)$, $Ω$ 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 $Ω\mapsto λ_{s,p}(Ω)$ under domain perturbations. In the case where $λ_{s,p}(Ω)$ 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 $Ω$. Thanks to the formula for our one-sided shape derivative, we characterize smooth local minimizers of $λ_{s,p}(Ω)$ 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 $λ_{s,p}(Ω)$ 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 $λ_{s,p}(B\setminus \overline B')$ is maximal when the two balls are concentric.

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