We present an abstract framework for treating the theory of wellposedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data.

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...

We consider existence and uniqueness for several examples of linear parabolic equations formulated on moving hypersurfaces. Specifically, we study in turn a surface heat equation, an equation posed on a bulk domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. In order to prove the well-posedness, we make use of an abstract framework presented in a recent work by the authors which dealt with the formulation and well-posedness of linear parabolic equations on arbitrary evolving Hilbert spaces. Here, after recalling all of the necessary concepts and theorems, we show that the abstract framework can applied to the case of evolving (or moving) hypersurfaces, and then we demonstrate the utility of the framework to the aforementioned problems.

One contribution of 15 to a theme issue 'Free boundary problems and related topics' . We formulate a Stefan problem on an evolving hypersurface and study the well posedness of weak solutions given L 1 data. To do this, we first develop function spaces and results to handle equations on evolving surfaces in order to give a natural treatment of the problem. Then, we consider the existence of solutions for L ∞ data; this is done by regularization of the nonlinearity. The regularized problem is solved by a fixed point theorem and then uniform estimates are obtained in order to pass to the limit. By using a duality method, we show continuous dependence, which allows us to extend the results to L 1 data.