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

## TL;DR

This paper investigates the obstacle problem for the Kolmogorov-Fokker-Planck operator with irregular coefficients, establishing existence, uniqueness, and variational inequalities using a variational approach in anisotropic Sobolev spaces.

## Contribution

It introduces a variational framework for the obstacle problem with rough coefficients and proves key properties like existence and uniqueness of solutions.

## Key findings

- Existence and uniqueness of weak solutions established.
- Development of anisotropic Sobolev space framework.
- Formulation of a variational inequality for the problem.

## Abstract

This work is devoted to the study of the obstacle problem associated to the Kolmogorov-Fokker-Planck operator with rough coefficients through a variational approach. In particular, after the introduction of a proper anisotropic Sobolev space and related properties, we prove the existence and uniqueness of a weak solution for the obstacle problem by adapting a classical perturbation argument to the convex functional associated to the case of our interest. Finally, we conclude this work by providing a one-sided associated variational inequality, alongside with an overview on related open problems.