Many phenomena in natural sciences are modelled by elliptic and parabolic partial differential equations (PDE). The most reknown examples of such PDE are the Laplace as well as the heat equation.
Once the solvability of these equations has been established, subsequent questions concern regularity properties of the solution u depending on the regularity of the data. In irregular situations, e.g., when these equations (complemented with suitable boundary conditions) are studied in a domain with a nonsmooth boundary the solution can develop singularities. In this talk, we will introduce to the challenges and problems arising in the study of elliptic and parabolic PDE in irregular situations. Some introductory examples from the context of the Laplacian will show us which regularity properties of solutions could possibly be expected. Afterwards, we will turn our focus to the Stokes equations and see some regularity results that can be established there.
