Title:	Second-order derivative of domain-dependent functionals along Nehari manifold trajectories
Authors:	Bobkov, Vladimír Kolonitskii, Sergey
Citation:	BOBKOV, V., KOLONITSKII, S. Second-order derivative of domain-dependent functionals along Nehari manifold trajectories. ESAIM-Control optimisation and calculus of variations, 2020, roč. 26, č. 48, s. 1-29. ISSN 1292-8119.
Issue Date:	2020
Publisher:	EDP Sciences
Document type:	článek article
URI:	2-s2.0-85091818874 http://hdl.handle.net/11025/39877
ISSN:	1292-8119
Keywords in different language:	Shape Hessian;second-order shape derivative;domain derivative;Hadamard formula;perturbation of boundary;superlinear nonlinearity;Nehari manifold;least energy solution;first eigenvalue
Abstract in different language:	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.
Rights:	© EDP Sciences Plný text není přístupný.
Appears in Collections:	Články / Articles (NTIS) OBD

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