Boundedness of weak solutions to degenerate Kolmogorov equations of hypoelliptic type in bounded domains

TL;DR

This paper proves the boundedness of weak solutions to a class of degenerate hypoelliptic Kolmogorov equations in bounded domains, extending classical results for parabolic equations using De Giorgi iteration and energy estimates.

Contribution

It extends boundedness results to degenerate hypoelliptic equations within bounded domains, employing a Lie group structure and novel energy estimates.

Findings

Established boundedness of weak subsolutions in bounded domains.

Extended classical boundedness theory to degenerate hypoelliptic equations.

Utilized energy estimates and the De Giorgi iteration method.

Abstract

We establish the boundedness of weak subsolutions for a class of degenerate Kolmogorov equations of hypoelliptic type, compatible with a homogeneous Lie group structure, within bounded product domains using the De Giorgi iteration. We employ the renormalization formula to handle boundary values and provide energy estimates. An $L^1$-$L^p$ type embedding estimate derived from the fundamental solution is utilized to incorporate lower-order divergence terms. This work naturally extends the boundedness theory for uniformly parabolic equations, with matching exponents for the coefficients.