31. Januar 2023
Dirk Pauly (Technische Universität Dresden)
Hilbert Complexes and PDEs
We present a simple way to solve linear PDEs using basic tools from linear functional analysis, so called FA-ToolBox. In this talk we focus on time-independent problems arising, e.g., in electro-magnetics, elas- ticity or general relativity. Hilbert complexes, adjoints, regular decompositions, and compact embeddings of certain domains of definition of unbounded linear operators will play a crucial role to prove compre- hensive solution theories together with all kinds of applications such as Friedrichs-Poincare type esti- mates, eigenvector expansions, div-curl lemmas, Hodge-Helmholtz decompositions, variational formula- tions, inf-sup lemmas, index of Dirac type operators, abstract trace spaces (quotient spaces/annihilators) and corresponding trace Hilbert complexes, ...
Our approach comprises bounded weak Lipschitz domains and mixed boundary conditions for which all presented Hilbert complexes are closed and even compact. Higher Sobolev order results are also shown. We shall point out connections and applications to algebra such a cohomology groups as well as to theo- retical numerics such as construction of finite elements or a posteriori error estimation for, e.g., FEM, BEM, or DEC.
In this lecture, among others, the most prominent examples of Hilbert complexes are the well-known de Rham complex for vector fields or differential forms, the elasticity complex, and the biharmonic complex.