Assume that a family of domain-dependent functionals $$ possesses a corresponding family of least energy critical points u$$ which can be found as (possibly nonunique) minimizers of $$ over the associated Nehari manifold $$. We obtain a formula for the second-order derivative of $$ with respect to t along Nehari manifold trajectories of the form $$, y ∈ Ω$$, where Φ$$ is a diffeomorphism such that Φ$$(Ω0) = Ω$$, α$$ ∈ ℝ is a $$-normalization coefficient, and v is a corrector function whose choice is fairly general. Since $$ is not necessarily twice differentiable with respect to t due to the possible nonuniqueness of u$$, 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 $$. 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.
Keywords: Shape Hessian, second-order shape derivative, domain derivative, Hadamard formula, perturbation of boundary, superlinear nonlinearity, Nehari manifold, least energy solution, first eigenvalue
@article{COCV_2020__26_1_A48_0,
author = {Bobkov, Vladimir and Kolonitskii, Sergey},
title = {Second-order derivative of domain-dependent functionals along {Nehari} manifold trajectories},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2019053},
mrnumber = {4144112},
zbl = {1453.35103},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2019053/}
}
TY - JOUR AU - Bobkov, Vladimir AU - Kolonitskii, Sergey TI - Second-order derivative of domain-dependent functionals along Nehari manifold trajectories JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2019053/ DO - 10.1051/cocv/2019053 LA - en ID - COCV_2020__26_1_A48_0 ER -
%0 Journal Article %A Bobkov, Vladimir %A Kolonitskii, Sergey %T Second-order derivative of domain-dependent functionals along Nehari manifold trajectories %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2019053/ %R 10.1051/cocv/2019053 %G en %F COCV_2020__26_1_A48_0
Bobkov, Vladimir; Kolonitskii, Sergey. Second-order derivative of domain-dependent functionals along Nehari manifold trajectories. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 48. doi: 10.1051/cocv/2019053
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