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

¹ Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia, Univerzitní 8, 301 00 Plzeň, Czech Republic.
² Institute of Mathematics, Ufa Federal Research Centre, RAS, Chernyshevsky str. 112, 450008 Ufa, Russia.
³ Saint Petersburg Electrotechnical University “LETI”, 5 Professora Popova st., St. Petersburg, 197376 Russia.

^* Corresponding author: bobkov@kma.zcu.cz

Received: 17 December 2018
Accepted: 28 August 2019

Abstract

Assume that a family of domain-dependent functionals E_Ωt possesses a corresponding family of least energy critical points u_t which can be found as (possibly nonunique) minimizers of E_Ωt over the associated Nehari manifold N(Ω_t). We obtain a formula for the second-order derivative of E_Ωt with respect to t along Nehari manifold trajectories of the form α_t(u₀(Φ_t⁻¹(y)) + tv(Φ_t⁻¹(y))), y ∈ Ω_t, where Φ_t is a diffeomorphism such that Φ_t(Ω₀) = Ω_t, α_t ∈ ℝ is a N(Ω_t)-normalization coefficient, and v is a corrector function whose choice is fairly general. Since E_Ωt [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_Ωt. 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.