MICS Seminar: Kaj Nystrom
Kaj Nystrom (Uppsala University) : Higher regularity of regular free boundaries for the Kolmogorov obstacle problem
Abstract
We establish higher regularity of regular free boundaries arising in the Kolmogorov obstacle problem. Under a quantitative thickness condition on the contact set, we prove that the free boundary is locally a non-characteristic hypersurface whose normal is Hölder continuous in the intrinsic Kolmogorov geometry. The proof combines the classification of blow-ups at regular points with boundary Harnack inequalities in asymptotically cylindrical Lipschitz domains. A central difficulty is that the derivatives determining the free-boundary normal do not satisfy a homogeneous Kolmogorov equation. We overcome this through a harmonic-replacement argument that controls the resulting error terms and yields Hölder continuity of the normal. Together with a decay estimate in the coupled transport-time direction, this leads to a full intrinsic improvement of flatness.
Biography
Kaj Nyström is Professor of Mathematical Analysis at Uppsala University. He received his PhD in mathematics from Umeå University in 1994 with a thesis entitled _Smoothness Properties of Solutions to Dirichlet Problems in Domains with a Fractal Boundary_. His research concerns partial differential equations, potential theory, harmonic analysis, and free-boundary problems, with particular emphasis on nonlinear, parabolic, and hypoelliptic equations.
