Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



A nonlinear history-dependent boundary value problem

Authors: Mircea Sofonea and Aissa Benseghir
Journal: Quart. Appl. Math. 75 (2017), 181-199
MSC (2010): Primary 35A01, 35Q74, 35R03, 47J20, 49J40
DOI: https://doi.org/10.1090/qam/1456
Published electronically: August 29, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a nonlinear boundary value problem with unilateral constraints in a two-dimensional rectangle. We derive a variational formulation of the problem which is in the form of a history-dependent variational inequality. Then, we establish the existence of a unique weak solution to the problem. We also prove two convergence results. The first one provides the continuous dependence of the solution with respect to the unilateral constraint. The second one shows the convergence of the solution of the penalized problem to the solution of the original problem, as the penalization parameter converges to zero.

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Additional Information

Mircea Sofonea
Affiliation: Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan Cedex, France
Email: sofonea@univ-perp.fr

Aissa Benseghir
Affiliation: Université Ferhat Abbas Sétif 1, El Bez Sétif 19000, Algérie
Email: aissa5919@yahoo.fr

DOI: https://doi.org/10.1090/qam/1456
Keywords: Nonlinear problem, unilateral constraint, history-dependent variational inequality, weak solution, penalty method, convergence results
Received by editor(s): March 3, 2016
Published electronically: August 29, 2016
Article copyright: © Copyright 2016 Brown University

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