Second-order derivative of domain-dependent functionals along Nehari manifold trajectories

Assume that a family of domain-dependent functionals $E_{{\mathrm{\Omega }}_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers of $E_{{\mathrm{\Omega }}_t}$ over the associated Nehari manifold $\mathcal{N}({\mathrm{\Omega }}_t)$. We obtain a formula for the second-order derivative of $E_{{\mathrm{\Omega }}_t}$ with respect to $t$ along Nehari manifold trajectories of the form $\alpha_t(u_0(\Phi_t^{-1}(y)) + t v (\Phi_t^{-1}(y)))$,$y \in \Omega_t$,where$\Phi_t$ is a diffeomorphism such that ${\mathrm{\Phi }}_t({\mathrm{\Omega }}_0)={\mathrm{\Omega }}_t$, ${\alpha }_t\in \mathbb{R}$ is a $\mathcal{N}({\mathrm{\Omega }}_t)$-normalization coefficient, and $v$ is a corrector function whose choice is fairly general. Since $E_{{\mathrm{\Omega }}_t}[u_t]$ is not necessarily twice differentiable with respect to $t$ due to the possible nonuniqueness of $u_t$, the obtained formula represents an upper bound for the corresponding second superdifferential, thereby providing a convenient way to study various domain optimization problems related to $E_{{\mathrm{\Omega }}_t}$. An analogous formula is also obtained for the first eigenvalue of the $p$-Laplacian. As an application of our results, we investigate the behaviour of the first eigenvalue of the Laplacian with respect to particular perturbations of rectangles.