Double phase problems with variable growth

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

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.

• Language: English
• Keywords: nonhomogeneous differential operator, double phase problem, high perturbation, spectrum of nonlinear operators
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2018
• Number of pages: Str. 270-287
• Numbering: Vol. 177, part A
• PID: 20.500.12556/RUL-109430
• UDC: 517.956
• ISSN on article: 0362-546X
• DOI: 10.1016/j.na.2018.03.016
• COBISS.SI-ID: 18353497
• Publication date in RUL: 02.09.2019

Record is a part of a journal

• Title: Nonlinear Analysis
• Shortened title: Nonlinear anal.
• Publisher: Pergamon Press
• ISSN: 0362-546X
• COBISS.SI-ID: 26027520

Secondary language

• Language: Slovenian
• Title: Problemi dvojne faze z variabilno rastjo

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