The aim is to develop asymptotic methods for dealing with nonsmooth domains when solving partial differential equations. The study of regularity properties of solutions is a central problem in PDE theory, and developing tools is very important since not much is known for nonsmooth domains. I also have a large interest in mathemical history of the more modern type (my master's thesis dealt with the history of continuous nowhere differentiable functions). Some of the topics I'm currently working on includes the following.
- Local estimates for Riesz potentials on nonsmooth surfaces.
- Local estimates for solutions to elliptic equations near bad points at the boundary; asymptotic methods.
- Hadamard type asymptotics for eigenvalues of elliptic operators under domain perturbations.
- Continuous nowhere differentiable functions (historical).