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We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted L 2 space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.

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...

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