A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

Litsgård, Malte; Nyström, Kaj

doi:10.2140/apde.2023.16.1547

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A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type

Malte Litsgård and Kaj Nyström

Vol. 16 (2023), No. 7, 1547–1588

DOI: 10.2140/apde.2023.16.1547

Abstract

We consider the operators

\nabla_{X} \cdot (A (X) \nabla_{X}), \nabla_{X} \cdot (A (X) \nabla_{X}) - \partial_{t}, \nabla_{X} \cdot (A (X) \nabla_{X}) + X \cdot \nabla_{Y} - \partial_{t},

where $X \in Ω$ , $(X, t) \in Ω \times ℝ$ and $(X, Y, t) \in Ω \times ℝ^{m} \times ℝ$ , respectively, and where $Ω \subset ℝ^{m}$ is an (unbounded) Lipschitz domain with defining function $ψ : ℝ^{m - 1} \to ℝ$ being Lipschitz with constant bounded by $M$ . Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure $d σ (X)$ and that the corresponding Radon–Nikodym derivative or Poisson kernel satisfies a scale-invariant reverse Hölder inequality in $L^{p}$ , for some fixed $p$ , $1 < p < \infty$ , with constants depending only on the constants of $A$ , $m$ and the Lipschitz constant of $ψ$ , $M$ . Under this assumption we prove that the same conclusions are also true for the parabolic measures associated to the second and third operators with $d σ (X)$ replaced by the surface measures $d σ (X) d t$ and $d σ (X) d Y d t$ , respectively. This structural theorem allows us to reprove several results previously established in the literature, as well as to deduce new results in, for example, the context of homogenization for operators of Kolmogorov type. Our proof of the structural theorem is based on recent results established by the authors concerning boundary Harnack inequalities for operators of Kolmogorov type in divergence form with bounded, measurable and uniformly elliptic coefficients.

Keywords

Kolmogorov equation, elliptic, parabolic, ultraparabolic, hypoelliptic, operators in divergence form, Dirichlet problem, Lipschitz domain, doubling measure, elliptic measure, parabolic measure, Kolmogorov measure, $A_{\infty}$, Lie group, homogenization

Mathematical Subject Classification

Primary: 35K65, 35K70, 35H20, 35R03

Milestones

Received: 14 December 2020

Revised: 15 December 2021

Accepted: 14 February 2022

Published: 21 September 2023

Authors

Malte Litsgård
Department of Mathematics Uppsala University Uppsala Sweden

Kaj Nyström
Department of Mathematics Uppsala University Uppsala Sweden

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Vol. 16, No. 7, 2023