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

Abstract

Assume that a family of domain-dependent functionals $E_{\Omega_t}$ possesses a corresponding family of least energy critical points $u_t$ which can be found as (possibly nonunique) minimizers of $E_{\Omega_t}$ over the associated Nehari manifold $\mathcal{N}(\Omega_t)$. We obtain a formula for the second-order derivative of $E_{\Omega_t}$ 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 $\Phi_t(\Omega_0) = \Omega_t$, $\alpha_t \in \mathbb{R}$ is a $\mathcal{N}(\Omega_t)$-normalization coefficient, and $v$ is a corrector function whose choice is fairly general. Since $E_{\Omega_t}[u_t]$ 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 $E_{\Omega_t}$. 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.