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*Erwin Schrödinger International Institute for Mathematical Physics*(Vienna, Austria).

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Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.

The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

*weak Riemannian*structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.

*right invariant*metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.

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