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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23520%2F20%3A43959290" target="_blank" >RIV/49777513:23520/20:43959290 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.esaim-cocv.org/articles/cocv/abs/2020/01/cocv180187/cocv180187.html" target="_blank" >https://www.esaim-cocv.org/articles/cocv/abs/2020/01/cocv180187/cocv180187.html</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1051/cocv/2019053" target="_blank" >10.1051/cocv/2019053</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS

ISSN

1292-8119

e-ISSN

—

Svazek periodika

26

Číslo periodika v rámci svazku

48

Stát vydavatele periodika

FR - Francouzská republika

Počet stran výsledku

29

Strana od-do

1-29

Kód UT WoS článku

000568562000005

EID výsledku v databázi Scopus

2-s2.0-85091818874