Matija Cencelj (Author), Vicenţiu Rǎdulescu (Author), Dušan Repovš (Author)

Abstract

Obravnavamo nek razred variacijskih integralov dvojne faze, ki se izrazijo z nehomogenimi potenciali. Proučujemo prirejeno Eulerjevo enačbo in izpostavimo obstoj dveh različnih Rayleighovih kvocientov. Eden izmed njih je v zvezi z obstojem neskončnega intervala lastnih vrednosti, medtem ko je drugi povezan z neobstojem lastnih vrednosti. Pojem lastne vrednosti razumemo v smislu parov nelinearnih operatorjev, kar so vpeljali Fučík, Nečas, Souček in Souček. V tem članku razširimo abstraktni okvir, ki ustreza nekaterim standardnim primerom povezanim s ▫$p(x)$▫-Laplaceovim operatorjem, posplošenim operatorjem povprečne ukrivljenosti ali pa diferencialnemu operatorju kapilarnosti z variabilnim eksponentom. Naši rezultati dopolnjujejo pionirske prispevke Marcellinija, Mingioneja in drugih na področju variacijskih integralov z neuravnovešeno rastjo.

Keywords

nehomogeni diferencial operator;problem dvojne faze;visoka perturbacija;spekter nelinearnih operatorjev;nonhomogeneous differential operator;double phase problem;high perturbation;spectrum of nonlinear operators;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 517.956
COBISS: 18353497 Link will open in a new window
ISSN: 0362-546X
Views: 523
Downloads: 380
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Other data

Secondary language: Slovenian
Secondary title: Problemi dvojne faze z variabilno rastjo
Secondary abstract: We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fučík, Nečas, Souček, and Souček. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the ▫$p(x)$▫-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.
Secondary keywords: nehomogeni diferencial operator;problem dvojne faze;visoka perturbacija;spekter nelinearnih operatorjev;
Type (COBISS): Article
Pages: str. 270-287
Volume: ǂVol. ǂ177
Issue: ǂpart ǂA
Chronology: Dec. 2018
DOI: 10.1016/j.na.2018.03.016
ID: 11213117