Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
Abstract
We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation 𝒦u=Δxu+x·∇yu−∂tu=0 in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable ym. Earlier theory required independence of ym. We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant C0,1/3ym defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for C0,αym graphs with α>1/3, as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without ym-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in ym. The resulting finite-dimensional translation representation yields real-analytic dependence on ym. Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent 1/3 is critical because ym has homogeneous degree three.
Related
- Weighted Kolmogorov equations: transfer estimates, local boundedness, and Harnack inequalities
- Fixed points for uniformly asymptotically Hölder-Lipschitz mappings
- Electrostatic inequalities and stability of matter on asymptotically Euclidean manifolds
- On uniqueness of probability solutions to one-dimensional Fokker-Planck-Kolmogorov equations
- Consistency in fractional Korn's inequalities on Lipschitz domains
BibTeX
@misc{nystrom2026boundary,
title = {{Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains}},
author = {Kaj Nyström},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29813},
eprint = {2608.29813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29813},
}
RIS
TY - UNPB
TI - Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
AU - Kaj Nyström
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29813
UR - https://arxiv.org/abs/2608.29813
AB - We establish boundary Harnack inequalities for non-negative solutions of the Kolmogorov equation 𝒦u=Δ_xu+x·∇_yu−∂_tu=0 in non-characteristic intrinsic Lipschitz graph domains whose defining functions may depend on the higher-order variable y_m. Earlier theory required independence of y_m. We replace this symmetry by quantitative asymptotic cylindricality, measured by a scale-invariant C^0,1/3_y_m defect. The perturbative comparison theorem applies whenever this defect is sufficiently small, and hence at all sufficiently small scales for C^0,α_y_m graphs with α>1/3, as well as under a vanishing critical cylindrical modulus. We obtain local comparability and intrinsic Hölder continuity of quotients of positive solutions, together with a Hölder estimate for their logarithmic quotient. The proof combines local boundary estimates, valid without y_m-independence, with a global rigidity argument for cylindrical blow-up limits. The central issue is uniform balance of forward and backward reference values. If this balance fails at collapsing scales, maximal-scale selection and a second rescaling produce a non-negative polynomial-growth Dirichlet solution in an unbounded cylindrical domain that vanishes at the backward reference point but not at the forward one. We exclude this configuration using boundary-energy and mean-value estimates, finite-dimensionality of spaces of polynomial-growth solutions, and translation invariance in y_m. The resulting finite-dimensional translation representation yields real-analytic dependence on y_m. Propagation of zeros, analytic continuation along invariant fibres, and a Tikhonov uniqueness argument then force the limiting solution to vanish identically. The exponent 1/3 is critical because y_m has homogeneous degree three.
ER -