ABSTRACT
This is a peculiar chapter that covers examples, extensions, and applications that do not seem to fit anywhere else in the layout of this book. We start with a detailed example of what has become known as T-coercivity. This is a sort of hidden ellipticity, which we show in a dual formulation of the Helmholtz equation, and in a diffusion problem with sign changing coefficient, loosely related to models of metamaterials. Next, we introduce some tools from differential calculus on normed spaces with the excuse of studying the dependence of diffusion problems with respect to the diffusion coefficient. From here, we move to studying elliptic problems and their interaction with convex optimization tools in Hilbert spaces, by looking for solutions on convex subsets of Sobolev spaces (the obstacle problem and the Signorini contact problem), by minimizing a functional that looks for the best data to match a given solution (a control problem) or by minimizing a nonquadratic convex function (friction boundary conditions). Finally, we finally leave the world of bounded operators and focus on a view of the Laplacian (and related operators) as an unbounded operator. Some of the natural properties that follow from this point of view lead into a black-box use of the theory of evolutionary equations in Hilbert spaces, which will give very simple while general results on the heat and wave equations.
