Elliptic Hemivariational Inequalities with Nonhomogeneous Neumann Boundary Conditions and Their Applications to Static F
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Elliptic Hemivariational Inequalities with Nonhomogeneous Neumann Boundary Conditions and Their Applications to Static Frictional Contact Problems Yong Huang · Zhenhai Liu · Stanislaw Migórski
Received: 5 January 2014 / Accepted: 16 July 2014 © Springer Science+Business Media Dordrecht 2014
Abstract This paper is devoted to the existence of solutions for variational–hemivariational inequalities of elliptic type with nonhomogeneous Neumann boundary conditions at resonance as well as at nonresonance. Using the notion of Clarke’s generalized gradient and the property of the first eigenfunction, we also build a Landesman-Lazer theory in the nonsmooth framework of variational–hemivariational inequalities of elliptic type. An application to a static frictional contact problem is provided. Keywords Boundary elliptic hemivariational inequality · Generalized Clarke subdifferential · Pseudomonotone operator · Existence of solutions · Frictional contact problem Mathematics Subject Classification (2000) 35B34 · 47J20 · 49J40 · 65N25
Project supported by NNSF of China Grants Nos. 11271087, 61263006, NSF of Guangxi Grant No. 2013GXNSFAA019022. The research was also supported by the Marie Curie International Research Staff Exchange Scheme Fellowship within the 7th European Community Framework Programme under Grant Agreement No. 295118 and the National Science Center of Poland under Maestro Advanced Project No. UMO-2012/06/A/ST1/00262. Y. Huang Department of Mathematics, Baise University, Baise 533000, Guangxi Province, P.R. China e-mail: [email protected]
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Z. Liu ( ) Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of Sciences, Guangxi University for Nationalities, Nanning 530006, Guangxi Province, P.R. China e-mail: [email protected] S. Migórski Jagiellonian University, Faculty of Mathematics and Computer Science, Institute of Computer Science, ul. Lojasiewicza 6, 30-348 Krakow, Poland e-mail: [email protected]
Y. Huang et al.
1 Introduction ∂Ω, W 1,p(Ω) with 1 < Let Ω ⊂ RN be an open bounded domain with a smooth boundary p < +∞, be the usual Sobolev space with the norm u = { Ω |∇u|p dx + Ω |u|p dx}1/p . For a convenience, in the following, we denote W 1,p (Ω) by V . Let A be a mapping from V to its dual space V ∗ which is defined by Au, vV =
N i=1
Ai (x, u, ∇u)Di v dx + Ω
A0 (x, u, ∇u)v dx
for all u, v ∈ V ,
(1)
Ω
where ·, ·V denotes the duality pairing between V ∗ and V , ∇ = (D1 , . . . , DN ), Di = ∂x∂ i and the functions Ai , for i = 0, 1, . . . , N satisfy suitable regularity and growth assumptions (see below). Let K be a nonempty, closed and convex subset of the Banach space V . In order to simplify computations we shall assume that 0 ∈ K. For any Banach space B, · B stands for the norm and ·, ·B denotes the associated duality pairing. The norm convergence in B and B ∗ is denoted by →. The weak convergence in B and B ∗ is denoted by . We formulate the following variational–hemivariational inequality denoted by (HVI): fin
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