Directional differentiability for elliptic quasi-variational inequalities of obstacle type

Abstract

The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments. arXiv:1802.03564v2 [math.OC] 27 Nov 2019 Remark 1.1. A space V under all of the previous assumptions except the second density assumption in (1) is referred to by Mignot in [49] as a 'Dirichlet space' -this is rather inconsistent with the modern literature [24] where Dirichlet spaces and Dirichlet forms are defined differently (see [24, §1.1]), for example in place of the T-monotonicity property the following Markov property should hold: if u ∈ V thenū := min(u + , 1) ∈ V and a(ū,ū) ≤ a(u, u). However, note that if the Markov property holds then so does T-monotonicity [48, Remark 1.4] [1, Proposition 5] and hence a Dirichlet form is also a positivity preserving form. Remark 1.2. It should be possible to generalise the above setting of positivity preserving spaces (V, a) to reflexive Banach spaces using for example the theory in [36] which generalises [49].Set H := L 2 (X; ξ) and let Φ : H → V be a possibly nonlinear map with Φ(0) ≥ 0 a.e., and suppose that it is increasing in the sense that u ≥ v a.e. (for u, v ∈ H) implies Φ(u) ≥ Φ(v) a.e. We define the set-valued mapping K :For fixed ϕ ∈ H, K(ϕ) is a closed, convex and non-empty set. Given f ∈ V * , consider the QVIIn general, there are multiple solutions of (P QVI ) (this and the existence theory for the QVI will be discussed later on) so we denote by Q : V * ⇒ V the set-valued mapping that takes a source term into the set of solutions of (P QVI ) with that right hand side source; hence (P QVI ) reads u ∈ Q(f ). When K is a constant mapping K(ϕ) ≡ K the problem (P QVI ) reduces to a standard variational inequality. In this work, we are interested in the differential sensitivity analysis of the map f → Q(f ); more precisely, we wish to show that a particular realisation of the multi-valued map Q is directionally differentiable. Such a result is of independent interest in itself but it is also a necessary step for deriving first order stationarity conditions for optimal control problems where the state is related to the control through a Remark 1.3. In fact, (A1) can be weakened significantly by requiring Hadamard differentiability of Φ only around the point u, i.e., locally, as in assumptions (A4) and (A5).Remark 1.4 (Compactness vs. complete continuity). Recall that a compact operator maps b...

Elliptic quasi-variational inequalities under a smallness assumption: uniqueness, differential stability and optimal control

“…We will not cover the general situation of a set-valued mapping Q : V ⇒ V , but we restrict the treatment of (1.1) to the case in which Q(y) is a moving set, i.e., Q(y) = K + Φ(y) (1.2) for some non-empty, closed and convex subset K ⊂ V and Φ : V → V . It is well-known that QVIs have many important real-world applications, we refer exemplarily to Bensoussan, Lions, 1987;Prigozhin, 1996;Barrett, Prigozhin, 2013;Alphonse, Hintermüller, Rautenberg, 2019 and the references therein.…”

Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

“…The QVI was first defined in [7] and has since become a standard tool for the modeling of various equilibrium-type scenarios in the natural sciences. The resulting applications include game theory [25], solid and continuum mechanics [8,26,43,50], economics [30,31], probability theory [41], transportation [13,20,55], biology [24], and stationary problems in superconductivity, thermoplasticity, or electrostatics [3,27,28,44,53]. For further information, we refer the reader to the corresponding papers, the monographs [5,43,48], and the references therein.…”

Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities

“…In this situation, the fixed-point problem (F) is equivalent to a so-called obstacle-type quasi-variational inequality (QVI) in which the bound defining the admissible set depends implicitly on the problem solution. Variational inequalities with such a structure arise, for instance, in the areas of mechanics, superconductivity, and thermoforming, see [2,3,8,31,32] and the references therein. As a prototypical example, we mention the following elliptic quasi-variational inequality that emerges in impulse control and that was one of the first QVIs to be formulated when this problem class was introduced by Lions and Bensoussan in the nineteen-seventies, cf.…”

“…Despite these difficulties, there have been several contributions in the recent years that have tried to establish stability and directional differentiability results for obstacle-type quasi-variational inequalities and, by doing so, to lay the foundation for the study of optimal control problems governed by QVIs. We mention exemplarily [2,4,5], which establish the directional differentiability of the solution maps of elliptic and parabolic obstacle-type quasi-variational inequalities in signed (i.e., nonnegative or nonpositive) directions by means of an approximation argument and classical results on ordinary variational inequalities; [6], which proves the continuity of the minimal and maximal solution mappings of elliptic obstacle-type QVIs in the L 2 -spaces; and [37], which establishes the directional differentiability of the solution operators of elliptic QVIs in all directions under a smallness assumption on the obstacle mapping and by means of the results of [17]. Unfortunately, all of the above papers have in common that they require very restrictive and, at times, even unrealistic assumptions on the involved operators and quantities.…”

“…See, e.g., the conditions on the sign of the directions appearing in the derivatives in [2, Theorem 1], [4, Assumption 28], and [5,Theorems 3.6,4.4]; the assumptions on the image, the complete continuity, and the size of Φ and its derivatives in [2, Assumptions (A2), (A3), (A5)], [4, Assumptions 28,32,34], [5,Theorems 3.6,4.4], [6, Assumption 1], and [37,Assumption 3.1]; and the comments in [1, Remark 2], which emphasize that compactness assumptions on the obstacle map are a main bottleneck in the study of obstacle-type QVIs. We remark that all of these conditions in particular prevent the differentiability results of [2,5,37] from being applicable to the problem (1.1).…”