A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
A Tykhonov-type well-posedness concept for elliptic hemivariational inequalities Rong Hu, Mircea Sofonea and Yi-bin Xiao Abstract. In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod–spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations. Mathematics Subject Classification. 35M86, 47J40, 49J52, 74K10, 74M15. Keywords. Hemivariational inequality, Tykhonov well-posedness, Convergence results, Contact problem, Spring–rod system.
1. Introduction Everywhere in this paper, unless stated otherwise, (X, · X ) is a real Banach space, ·, · denotes the duality pairing between X and its dual X ∗ , K is a nonempty subset of X, A : X → X ∗ , j : X → R is a locally Lipschitz function and f ∈ X ∗ . We denote by j 0 (u; v) the generalized directional derivative of j at the point u in the direction v, see Definition 2. With these notation, we consider the hemivariational inequality u ∈ K,
Au, v − u + j 0 (u; v − u) ≥ f, v − u
∀ v ∈ K.
(1.1)
Such kind of inequalities arise in Contact Mechanics. They model the equilibrium of elastic bodies acted upon the body forces and surface tractions, in frictional or frictionless contact with an obstacle. References in the field are [15,16] and, more recently [2,13,18,22,28,29]. There, existence and uniqueness results for inequality problems of the form (1.1) can be found, under various assumptions on the data. A convergence result for such inequalities was provided in [30] and general results on their numerical analysis of such inequalities can be found in [4–6]. Results on the Tykhonov regularization for hemivariational inequalities can be found in the recent paper [23]. The current paper was inspired by three types of studies related to the hemivariational inequality (1.1): the well-posedness in the sense of Tykhonov, the continuous dependence of the solution with respect to the data, and the perturbation with a convex function, under specific assumptions. We briefly describe in what follows each of these approaches. First, the concept of well-posedness in the sense of Tykhonov was introduced for minimization problems in [24]. Later, it was extended to variational i
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