Bourgain–Spencer accuracy for random elliptic media at finite contrast
Let $d \ge 2$. Let $\xi$ be vector-valued Gaussian white noise on $\mathbb{R}^d$, let $C_0$ be a smooth compactly supported matrix-valued kernel, $G = C_0 * \xi$, and $a(x) = a_0(G(x))$, where $a_0 \in C^1_b$ takes symmetric matrix values with $\lambda I \le a_0 \le I$ for some $\lambda > 0$. For deterministic $f \in C^\infty_c(\mathbb{R}^d; \mathbb{R}^d)$ and $\varepsilon > 0$ let $\nabla u_\varepsilon \in L^2$ be the unique gradient solution of $$-\nabla \cdot a(x/\varepsilon) \nabla u_\varepsilon = \nabla \cdot f.$$ Given constant tensors $A^1, \dots, A^{2d}$, where each $A^k_{j_1 \cdots j_{k-1}}$ is a $d \times d$ matrix and $A^1$ is positive definite, define deterministic gradients $\nabla v_n$ in $L^2$ recursively by $-\nabla \cdot A^1 \nabla v_1 = \nabla \cdot f$ and $$-\nabla \cdot A^1 \nabla v_n = \nabla \cdot \sum_{k=2}^{n} A^k_{j_1 \cdots j_{k-1}} \partial_{j_1} \cdots \partial_{j_{k-1}} \nabla v_{n+1-k}, \qquad 2 \le n \le 2d,$$ with repeated indices summed, and set $U_\varepsilon = \sum_{n=1}^{2d} \varepsilon^{n-1} v_n$, a proxy for the solution of the higher-order homogenized equation.
For a discrete analogue with i.i.d. coefficients within $\delta$ of a constant matrix, Bourgain's result, refined by Kim and Lemm, implies that $\mathbb{E} \nabla u_\varepsilon$ admits such a description to order $\varepsilon^{2d - C\delta}$; Duerinckx, Lemm and Pagano extended this to continuum coefficient fields, again for small $\delta$. Without a smallness assumption, Duerinckx proved accuracy $\varepsilon^{d - \eta}$ for Gaussian coefficient fields, measured in a norm that is $L^2$ on scale $\varepsilon$ and $L^p$, with $p$ large, on large scales.
Problem. Do there exist constant tensors $A^1, \dots, A^{2d}$, depending only on the law of $a$, with $A^1$ positive definite, such that $$\|\mathbb{E} \nabla u_\varepsilon - \nabla U_\varepsilon\|_{L^2(\mathbb{R}^d)} \le C_{f,\eta}\, \varepsilon^{2d - \eta} \qquad (0 < \varepsilon \le 1)$$ for every $f$ and every $0 < \eta < 1$? No small-contrast assumption is made on $a$; the constants may depend on its law and ellipticity.
Source: Bourgain and Spencer, after Bourgain (2018) and Kim and Lemm (2019); named the Bourgain–Spencer conjecture by Duerinckx, Gloria and Lemm (2019); in the form of Duerinckx (2023), Conjecture 1. Stated by AIM — Open Applied Problems for finite-range Gaussian coefficient fields.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- PR
- Keywords
- AIM Open Applied Problems, higher-order homogenized equations, stochastic homogenization
References
@article{duerinckx2023,
author = {Duerinckx, Mitia},
title = {Non-perturbative approach to the {B}ourgain--{S}pencer conjecture in stochastic homogenization},
journal = {J. Math. Pures Appl.},
volume = {176},
year = {2023},
pages = {183--225},
doi = {10.1016/j.matpur.2023.06.005},
eprint = {2102.06319},
archivePrefix = {arXiv},
}
@article{duerinckx2019,
author = {Duerinckx, Mitia and Gloria, Antoine and Lemm, Marius},
title = {A remark on a surprising result by {B}ourgain in homogenization},
journal = {Comm. Partial Differential Equations},
volume = {44},
number = {12},
year = {2019},
pages = {1345--1357},
doi = {10.1080/03605302.2019.1638934},
eprint = {1903.05247},
archivePrefix = {arXiv},
}
@article{bourgain2018,
author = {Bourgain, Jean},
title = {On a homogenization problem},
journal = {J. Stat. Phys.},
volume = {172},
number = {2},
year = {2018},
pages = {314--320},
doi = {10.1007/s10955-018-1981-5},
}
@article{kim2019,
author = {Kim, Jongchon and Lemm, Marius},
title = {On the averaged {G}reen's function of an elliptic equation with random coefficients},
journal = {Arch. Ration. Mech. Anal.},
volume = {234},
number = {3},
year = {2019},
pages = {1121--1166},
doi = {10.1007/s00205-019-01409-1},
}
@article{duerinckx2025,
author = {Duerinckx, Mitia and Lemm, Marius and Pagano, Fran{\c{c}}ois},
title = {On {B}ourgain's approach to stochastic homogenization},
journal = {Arch. Ration. Mech. Anal.},
volume = {249},
number = {6},
year = {2025},
pages = {81},
doi = {10.1007/s00205-025-02145-5},
eprint = {2406.09909},
archivePrefix = {arXiv},
}
@misc{aim-bourgain-spencer-nonperturbative-homogenization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Bourgain–Spencer accuracy for random elliptic media at finite contrast}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/453-bourgain-spencer-nonperturbative-homogenization.md},
note = {Entry 453 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00412, "Bourgain–Spencer accuracy for random elliptic media at finite contrast", clopen.solutions.