On the fundamental solution for degenerate Kolmogorov equations with rough coefficients

Abstract

The aim of this work is to prove the existence of a fundamental solution associated to the Kolmogorov equation $$\mathscr {L}u = f$$ L u = f in the dilation invariant case, with bounded measurable first order coefficients and bounded diffusion coefficients satisfying a sort of divergence free assumption. Finally, we prove Gaussian upper and lower bounds for the fundamental solution, and other related properties, under less restrictive assumptions on the coefficients.

On the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients

“…The extension of the De Giorgi-Nash-Moser regularity theory to this setting had been an open problem for decades recently resolved both in the kinetic and ultraparabolic setting, with various contributions among which we recall [15,16,17] and [4], respectively. As far as we are concerned with well-posedness results for boundary value problems in the weak setting, there are recent results regarding existence and uniqueness of the solution for the Dirichlet problem [24], existence of a weak fundamental solution for the weak Cauchy problem [5], and finally C α regularity estimates up to the boundary [33,31]. It is in this weak framework that we aim to address the study of the weak obstacle problem by means of variational methods, a topic which presents many interesting open problems that we discuss in Section 4.…”