Usually the inverse mean curvature flow (IMCF) of convex hypersurfaces exists for all time and flows outwards towards infinitely large spheres. In this talk we will consider IMCF on of convex disks inside the unit ball with a Neumann boundary condition on the sphere. We will see that conversely to the boundaryless case, global singularities occur in finite time, and the flow converges in C^{1,\beta} to a flat disk.

I will illustrate recent developments of the theory of Sobolev spaces on metric measure spaces. In particular I will focus on the more recent viewpoint, inspired by Weaver’s derivations, which provides an integration by parts formula even in the abstract setting. I will show how this viewpoint is completely equivalent to the usual one constructed by approximation with Lipschitz functions under very mild assumptions on the metric measure structure. If there is time I will discuss also the theory for BV and $W^{1,1}$ spaces.