Minimax Theory and its Applications 05 (2020), No. 1, 103--128
Copyright Heldermann Verlag 2020
A Garding Inequality Based Unified Approach to Various Classes of Semi-Coercive Variational Inequalities Applied to Non-Monotone Contact Problems with a Nested Max-Min Superpotential
Joachim Gwinner
Institut für Mathematik und Rechneranwendung, Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr, 85577 München-Neubiberg, Germany
joachim.gwinner@unibw-muenchen.de
Nina Ovcharova
Institut für Mathematik und Rechneranwendung, Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr, 85577 München-Neubiberg, Germany
We present a unified existence and approximation theory for various classes of variational inequalities (VIs) in reflexive Banach spaces. The focus is on semi-coercive problems. Here we abandon projections, which are limited to a Hilbert space setting, instead we adopt semicoercivity of the elliptic linear operator in form of a Garding inequality. Also we extend the smoothing procedure from \cite{Ov-Gw-14} to provide smoothing approximations of nested max-min functions. Then we couple this regularization technique with the finite element method to solve numerically semi-coercive hemivariational inequalities (HVIs) involving a nested max-min superpotential and apply our approximation theory for pseudomonotone VIs to these HVIs. As a model example we consider a unilateral semi-coercive contact problem with non-monotone friction on the contact boundary.
Keywords: Hemivariational inequality, pseudomonotonicity, semicoercivity, Garding inequality, max-min superpotential, smoothing approximation, finite element discretization, non-monotone contact.
MSC: 35J65, 35J86, 35J87, 49J40, 74M10, 74M15, 74S05.
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