Second variation of the fractional torsion energy and linear stability of the ball

We compute the second shape derivative of the fractional torsion. It is an explicit quadratic form on the boundary, which involves a Dirichlet-to-Neumann operator for half-harmonic functions blowing up at the boundary. We deduce an explicit formula for volume preserving variations on the ball and show that as in the local case, the shape Hessian is negative and coercive in \(H^{1/2}\) modulo translations.