Volume 26, 2020
|Number of page(s)||29|
|Published online||03 September 2020|
Second-order derivative of domain-dependent functionals along Nehari manifold trajectories
Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia,
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: firstname.lastname@example.org
Accepted: 28 August 2019
Assume that a family of domain-dependent functionals EΩt possesses a corresponding family of least energy critical points ut which can be found as (possibly nonunique) minimizers of EΩt over the associated Nehari manifold N(Ωt). We obtain a formula for the second-order derivative of EΩt with respect to t along Nehari manifold trajectories of the form αt(u0(Φt−1(y)) + tv(Φt−1(y))), y ∈ Ωt, where Φt is a diffeomorphism such that Φt(Ω0) = Ωt, αt ∈ ℝ is a N(Ωt)-normalization coefficient, and v is a corrector function whose choice is fairly general. Since EΩt [ut] is not necessarily twice differentiable with respect to t due to the possible nonuniqueness of ut, 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 EΩt. 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.
Mathematics Subject Classification: 35J92 / 49Q10 / 35B30 / 49K30
Key words: Shape Hessian / second-order shape derivative / domain derivative / Hadamard formula / perturbation of boundary / superlinear nonlinearity / Nehari manifold / least energy solution / first eigenvalue
© EDP Sciences, SMAI 2020
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