Partial differential equations in generalized Sobolev spaces
Final Report Abstract
In this joint German-Ukrainian project, the solvability and well-posedness of a large class of partial differential equations are studied. More precisely, we investigate elliptic equations in bounded domains with appropriate conditions on the boundary of the domain - this class of problems appearing in many applications from physics and engineering. Under standard assumptions, one can prove regularity properties of the solution and uniform estimates, showing continuous dependence of the solution on the data. While elliptic theory is a classical field in partial differential equations, the novelty of this project lies in the choice of the spaces for the solution and the data: we consider so-called Hörmander spaces, which form a refined scale of solution spaces compared to the more classical choice of Sobolev spaces. The focus of this project lies in Hormander spaces of low (and negative) regularity. In this way, one can include irregular source terms and random effects, in particular noise terms which affect the system on the boundary. The connection between Hörmander spaces and the regularity of white noise was surprising and could lead to a better understanding of the paths of white noise. This project also gives some contribution to the interpolation of Hilbert spaces, an abstract method from functional analysis which is useful in the investigation of partial differential equations.
Publications
- Elliptic problems with rough boundary data in generalized Sobolev spaces. 40 pages
Anna Anop, Robert Denk, Alexandr Murach
