Extremal solutions of multi-valued variational inequalities in plane exterior domains

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:267

Article

Carl, Siegfried¹  Le, Vy K.² 

[1] Martin Luther Univ Halle Wittenberg, Inst Math, D-06099 Halle, Germany

[2] Missouri Univ Sci & Technol, Dept Math & Stat, Rolla, MO 65409 USA

关键词: Variational inequality;    Exterior plane domain;    Extremal solution;    Multivalued mapping;    Variational-hemivariational inequalities;   

DOI  :  10.1016/j.jde.2019.05.020

来源: Elsevier

【 摘 要 】

Let Omega =R-2\(B(0, 1))over bar> be the exterior of the closed unit disc in the plane. In this paper we prove existence and enclosure results of multi-valued variational inequalities in Omega of the form: Find u is an element of K and eta is an element of F(u) such that < - Au, v - u > >= < a eta, v - u >, for all v is an element of K , where K is a closed convex subset of the Hilbert space X= D-0(1,2)(Omega) which is the completion of C-c(infinity)(Omega) with respect to the vertical bar vertical bar del . vertical bar vertical bar 2,Omega-norm. The lower order multi-valued operator F is generated by an upper semicontinuous multi-valued function integral : R -> 2(R)\{phi}, and the ( single-valued) coefficient a : Omega -> R+ is supposed to decay like |x|(-2-alpha) with alpha > 0. Unlike in the situation of higher-dimensional exterior domain, that is R-N\ B( 0,1) with N >= 3, the borderline case N = 2 considered here requires new tools for its treatment and results in a qualitatively different behaviour of its solutions. We establish a sub-supersolution principle for the above multi-valued variational inequality and prove the existence of extremal solutions. Moreover, we are going to show that classes of generalized variational-hemivariational inequalities turn out to be merely special cases of the above multi-valued variational inequality. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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