Abstract for the talk on 10.01.2023 (10:00 h)
VortragAnnalaura Rebucci (Università degli Studi di Modena e Reggio Emilia, Italy)
Regularity results and new perspectives for degenerate Kolmogorov equations
We are concerned with the regularity theory of strongly degenerate Kolmogorov equations and we also study a relativistic generalization of such equations. We divide
this presentation into three parts.
In the first part, we present some results which lie in the classical regularity theory of Kolmogorov-type operators with regular coefficients. In particular, we here discuss some Schauder estimates for classical solutions to Kolmogorov equations in non-divergence form with Dini-continuous coefficients contained in. Furthermore, we show new pointwise regularity results and a Taylor-type expansion up to second order
with estimate of the rest in Lp norm.
The second part focuses on the weak regularity theory of degenerate Kolmogorov equations with discontinuous coefficients, which is nowadays the main focus of the research community. More precisely, we present a Harnack inequality and the H ̈older continuity
for weak solutions to the Kolmogorov equation with measurable coefficients, integrable lower order terms and nonzero source term, following the work.
Finally, in the last part of this presentation, we address a possible generalization of the kinetic Kolmogorov-Fokker-Planck equation, which is in accordance with the theory of special relativity and was studied in. In particular, we explain why the operator proposed is the suitable relativistic generalization of the Fokker-Planck operator and we describe it as a H ̈ormander operator which is invariant with respect to Lorentz transformations.