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 ${\alpha }_t(u_0({\Phi }_t^{-1}(y))+tv({\Phi }_t^{-1}(y)))$, $y\in {\Omega }_t$, 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.

25 pages, 6 figures. The title has been updated and the exposition has been improved according to the referee's suggestions. Accepted to ESAIM: Control, Optimisation and Calculus of Variations