Caffarelli Estimates under Lipschitz Perturbations
Abstract
We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let d≥1, L≥0, and let B:ℝd→ℝ be globally L-Lipschitz. We prove that the Brenier map from the standard Gaussian measure γd to the probability measure proportional to e−Bγd has a globally Lipschitz representative whose bound depends only on L. The Brenier potential also belongs to C1,1(ℝd) and satisfies dimension-free two-sided Hessian bounds. Let C(0,L) denote the upper-bound constant, then logC(0,L)=4L2+log L+𝒪(1) (L→∞), and a one-dimensional example with Brenier map TL satisfies logLip(TL)≥ L2/2. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let V,W:ℝd→ℝ be potentials, and let Q,P be positive-definite matrices. Using the distributional curvature bounds D2V⪯ Q and D2W⪰ P, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.
Related
- Singular set estimates in quasilinear Alt-Caffarelli problems
- Maximal Monotone Differential Inclusions with Volterra and One-Sided Lipschitz Perturbations under Nonlocal Initial Conditions and Applications
- Gradient estimates for the Lagrangian mean curvature equation with Lipschitz critical phase
- Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides
- Resolvent Estimates for the Stokes Operator in a Three-Dimensional Lipschitz Domain
BibTeX
@misc{gwozdz2026caffarelli,
title = {{Caffarelli Estimates under Lipschitz Perturbations}},
author = {Maja Gwóźdź},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04052},
eprint = {2609.04052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04052},
}
RIS
TY - UNPB
TI - Caffarelli Estimates under Lipschitz Perturbations
AU - Maja Gwóźdź
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04052
UR - https://arxiv.org/abs/2609.04052
AB - We study dimension-free differential estimates for Brenier maps under first-order perturbations of the two marginals. Caffarelli's contraction theorem yields such estimates when the source and target potentials satisfy a pointwise Hessian comparison. We show that the same conclusion remains true under arbitrary globally Lipschitz perturbations of both marginals, without additional assumptions. In particular, this answers the conjecture due to Fathi, Mikulincer, and Shenfeld. More precisely, let d≥1, L≥0, and let B:ℝ^d→ℝ be globally L-Lipschitz. We prove that the Brenier map from the standard Gaussian measure γ_d to the probability measure proportional to e^−Bγ_d has a globally Lipschitz representative whose bound depends only on L. The Brenier potential also belongs to C^1,1(ℝ^d) and satisfies dimension-free two-sided Hessian bounds. Let C(0,L) denote the upper-bound constant, then logC(0,L)=4L^2+log L+𝒪(1) (L→∞), and a one-dimensional example with Brenier map T_L satisfies logLip(T_L)≥ L^2/2. We deduce this Gaussian estimate from an anisotropic two-marginal result. Let V,W:ℝ^d→ℝ be potentials, and let Q,P be positive-definite matrices. Using the distributional curvature bounds D^2V⪯ Q and D^2W⪰ P, we establish matrix Hessian bounds for Brenier maps between arbitrary globally Lipschitz perturbations of the reference marginals. Most importantly, the constants keep the directional geometry of the essential gradient ranges. For affine perturbations, our estimates recover the sharp noncommuting Caffarelli tensor. We further obtain pointwise displacement bounds for Gaussian perturbations and spectral Hessian estimates for compact Gaussian mixtures.
ER -