Well-posedness of the local-profile (defect corrector) equation for a periodic elliptic coefficient perturbed by a defect that merely vanishes at infinity: the $r=\infty$ case

For the elliptic equation $-\operatorname{div}(a(x/\varepsilon)\nabla u^\varepsilon)=f$ with an oscillatory coefficient $a=a_{\mathrm{per}}+b$ consisting of a periodic background perturbed by a localized defect $b$, the fine-scale behavior of $u^\varepsilon$ near the defect is governed by a local profile $w_p$: the solution, for each $p\in\mathbb{R}^d$, of the corrector-type equation $-\operatorname{div}(a(p+\nabla w_p))=0$ on $\mathbb{R}^d$ subject to strict sublinearity at infinity. Blanc, Le Bris and Lions established existence and uniqueness (up to additive constants) of such profiles for defects $b\in L^r(\mathbb{R}^d)$, $1\le r<+\infty$, but explicitly left open the natural and more general case — the 'ideal' framework of their program — in which $b$ is bounded and merely tends to zero at infinity, with no integrability assumed (the endpoint case $r=\infty$). This problem asks for a rigorous resolution of that case: either a well-posedness theorem (existence, uniqueness up to constants, and the decay and regularity of the profile perturbation) for bounded Hölder-continuous defects vanishing at infinity, or an explicit counterexample within this class showing that no strictly sublinear profile exists or that uniqueness fails. The same obstruction governs every extension of the program (non-divergence form, advection-diffusion, semilinear equations, systems), all of which are so far established only for $L^r$-type defects.

Background

Homogenization theory studies equations whose coefficients oscillate rapidly at a small scale $\varepsilon$, here the scalar divergence-form elliptic equation $-\operatorname{div}(a(x/\varepsilon)\nabla u^\varepsilon)=f$ on a bounded smooth domain $\Omega\subset\mathbb{R}^d$ with homogeneous Dirichlet conditions, $d\ge 3$. When $a=a_{\mathrm{per}}$ is periodic (say $\mathbb{Z}^d$-periodic), bounded and uniformly elliptic (i.e. $0<\lambda\le a\le\Lambda$), the solutions $u^\varepsilon$ converge to the solution $u^*$ of a homogenized equation $-\operatorname{div}(A^*\nabla u^*)=f$ with constant effective matrix $A^*$, and the fine-scale oscillations are described by the two-scale expansion $u^\varepsilon\approx u^*+\varepsilon\sum_i w^{e_i}(x/\varepsilon)\partial_i u^*$, where for each vector $p\in\mathbb{R}^d$ the periodic corrector $w_{p,\mathrm{per}}$ solves the corrector equation $-\operatorname{div}(a_{\mathrm{per}}(p+\nabla w_{p,\mathrm{per}}))=0$ on $\mathbb{R}^d$ with $w_{p,\mathrm{per}}$ periodic (hence sublinear at infinity, and unique up to an additive constant); see the classical monograph of Jikov, Kozlov and Oleinik.

This classical picture does not describe materials whose periodic microstructure carries a defect: a coefficient $a=a_{\mathrm{per}}+b$ where $b$ is a perturbation localized near the origin and decaying at infinity. The macroscopic (homogenized) behavior is unchanged, but near the defect the solution at scale $\varepsilon$ is modified, and capturing it requires the local profile $w_p$: the solution of the same corrector equation with the perturbed coefficient, $-\operatorname{div}(a(p+\nabla w_p))=0$ on $\mathbb{R}^d$, now posed on the whole space with only the sublinearity condition $w_p(x)/(1+|x|)\to 0$ as $|x|\to\infty$, and expected to decompose as $w_p=w_{p,\mathrm{per}}+\tilde w_p$ with $\tilde w_p$ small at infinity. Such a profile is the analogue, for a defect, of the periodic corrector, and its well-posedness is the cornerstone of the corresponding two-scale approximations.

Blanc, Le Bris and Lions established this well-posedness in stages. For $b\in L^2(\mathbb{R}^d)$ it follows from the Lax–Milgram theorem (the standard existence theorem for coercive variational problems) after a zeroth-order regularization. For $b\in L^r(\mathbb{R}^d)$ with $1\le r<d$, writing the equation for $\tilde w_p$ as $-\operatorname{div}(a\nabla\tilde w_p)=\operatorname{div}(b(p+\nabla w_{p,\mathrm{per}}))$ and representing the solution through the Green function (the fundamental solution of $-\operatorname{div}(a\nabla\cdot)$, i.e. the kernel $G(x,y)$ solving $-\operatorname{div}_x(a\nabla_x G)=\delta(x-y)$), Marcinkiewicz weak-$L^q$ estimates (bounds in the weak Lebesgue space $L^{q,\infty}$, the space of functions whose distribution functions decay like $t^{-q}$) yield existence of a bounded $\tilde w_p$ vanishing at infinity, and uniqueness up to constants follows from a Liouville-type theorem (bounded entire solutions of homogeneous elliptic equations with bounded coefficients are constant). For general finite $r$, in particular the delicate supercritical regime $r>d$, Green-function estimates on dyadic rings $\{R\le|x-y|\le 2R\}$, combined with the classical compactness and $L^p$ regularity estimates that Avellaneda and Lin established for elliptic equations with periodic coefficients, give a solution with $\nabla\tilde w_p\in L^r\cap L^\infty$ when the coefficients are Hölder continuous; the exponent $r=d$ is critical (heuristically, $b(y)\sim|y|^{-1}$ sits at the borderline of $L^d$).

All of these arguments consume the $L^r$ integrability of $b$ to sum the Green-function estimates over dyadic rings, and the linear operator mapping $b$ to $\nabla\tilde w_p$ is of Calderón–Zygmund type (a singular-integral operator with a periodically oscillating kernel): by the Avellaneda–Lin theory it is bounded on $L^r$ for $1<r<\infty$, but the case of a defect merely vanishing at infinity sits precisely at the $L^\infty$/$C_0$/BMO endpoints (BMO, the space of functions of bounded mean oscillation, being the natural substitute class at such endpoints), where boundedness of Calderón–Zygmund operators typically fails. The authors state the situation explicitly: 'the natural assumption would be $b(x)\to0$ as $|x|\to\infty$; however, we are unable to address the problem in this full generality', and 'We are unfortunately unable to conclude on the general case of a perturbation $b$ that only goes to zero at infinity, i.e. the case $r=\infty$.' The question is physically meaningful: strain and disorder fields around real defects decay algebraically with small exponents or even logarithmically; an algebraic tail $(1+|x|)^{-\beta}$ lies in $L^r(\mathbb{R}^d)$ once $r>d/\beta$, so it is the logarithmic (and slower) decay, belonging to no $L^r(\mathbb{R}^d)$, that lies beyond the cited finite-$r$ theorems. The question is also delicate: in a neighboring class of coefficients (almost translation-invariant coefficients, characterized by a discrete gradient — the vector of finite differences $g(\cdot+e_i)-g$ along the coordinate lattice directions — lying in $L^p$), Goudey showed that for $p\ge d$ homogenization can genuinely fail (different subsequences converge to different limits), so slow decay of perturbations is a real mechanism of breakdown rather than a mere technical annoyance. The question below is the 'most immediate open problem' of the program and remains open.

Problem Statement

Let $d\ge 3$ (the setting of the source; dimension 1 is explicitly solvable and dimension 2 requires separate care). Let $a_{\mathrm{per}}:\mathbb{R}^d\to\mathbb{R}$ be bounded, uniformly elliptic ($0<\lambda\le a_{\mathrm{per}}\le\Lambda$), $\mathbb{Z}^d$-periodic and of Hölder class $C^{0,\alpha}$ for some $\alpha\in(0,1)$ (i.e. $|a_{\mathrm{per}}(x)-a_{\mathrm{per}}(y)|\le C|x-y|^{\alpha}$ for a constant $C$ independent of $x$ and $y$), and let $b:\mathbb{R}^d\to\mathbb{R}$ be bounded and of class $C^{0,\alpha}$, such that $a:=a_{\mathrm{per}}+b$ is uniformly elliptic and

$$\lim_{|x|\to\infty} b(x)=0,$$

with no integrability assumption on $b$ — in particular $b$ need not belong to $L^r(\mathbb{R}^d)$ for any finite $r$ (bounded tails such as $b(x)\sim 1/\log(2+|x|)$ lie in no finite $L^r$, whereas an algebraic tail $b(x)\sim(1+|x|)^{-\beta}$ lies in $L^r$ once $r>d/\beta$ and is therefore already covered by the finite-$r$ theorems; the class contains both). For each $p\in\mathbb{R}^d$, consider the local-profile (defect corrector) equation

$$-\operatorname{div}\big(a\,(p+\nabla w_p)\big)=0 \quad\text{in } \mathbb{R}^d,$$

together with the strict sublinearity condition

$$\frac{w_p(x)}{1+|x|}\longrightarrow 0 \quad\text{as } |x|\to\infty.$$

The questions are: (1) Existence — does there exist a distributional solution of the form $w_p=w_{p,\mathrm{per}}+\tilde w_p$, where $w_{p,\mathrm{per}}$ is the periodic corrector of the unperturbed coefficient? (2) Uniqueness — is such a solution unique up to the addition of constants? (3) Qualitative structure — what decay, boundedness or substitute integrability (for instance in a BMO-type or uniformly local class) hold for $\tilde w_p$ and $\nabla\tilde w_p$, and are these properties sufficient for the two-scale approximation of $u^\varepsilon$ near the defect, as they are for $L^r$ defects?

Resolving this question means either proving well-posedness for the full class of perturbations above — affirmative answers to questions (1) and (2) together with the decay, boundedness or substitute integrability of $\tilde w_p$ and $\nabla\tilde w_p$ requested in the first part of question (3) — or exhibiting an explicit counterexample within that class: a bounded, Hölder-continuous, uniformly elliptic perturbation $b$ vanishing at infinity for which no strictly sublinear solution exists, or for which uniqueness up to constants fails. The second part of question (3), on whether those properties support the two-scale approximation of $u^\varepsilon$ near the defect, is a further question of the program. Results restricted to $b\in L^r(\mathbb{R}^d)$ for some $r<+\infty$ belong to the known theory rather than to this question. The identical $r=\infty$ obstruction governs the extensions of the program (non-divergence form, advection-diffusion, semilinear equations, systems), which are established in the literature only for $L^r$-type defects; the present problem is the divergence-form scalar case that the source singles out as the most immediate.

References

1. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'Profils locaux et problèmes elliptiques à plusieurs échelles avec défauts' (Local profiles and elliptic problems at different scales with defects), C. R. Acad. Sci. Paris, Ser. I 353 (2015) 203–208. DOI: 10.1016/j.crma.2015.01.003, https://doi.org/10.1016/j.crma.2015.01.003 (full text also at https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2015.01.003.pdf)
2. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'A possible homogenization approach for the numerical simulation of periodic microstructures with defects', Milan J. Math. 80 (2012) 351–367. DOI: 10.1007/s00032-012-0186-7, https://doi.org/10.1007/s00032-012-0186-7
3. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures', Commun. Partial Differ. Equ. 40 (2015) 2173–2236. DOI: 10.1080/03605302.2015.1043464, https://doi.org/10.1080/03605302.2015.1043464 (full text at https://hal.science/hal-01143193)
4. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'On correctors for linear elliptic homogenization in the presence of local defects', Commun. Partial Differ. Equ. 43 (2018) 965–997. DOI: 10.1080/03605302.2018.1484764, https://doi.org/10.1080/03605302.2018.1484764 (preprint arXiv:1801.10335, https://arxiv.org/abs/1801.10335)
5. Xavier Blanc, Claude Le Bris, Pierre-Louis Lions, 'On correctors for linear elliptic homogenization in the presence of local defects: the case of advection-diffusion', J. Math. Pures Appl. (9) 124 (2019) 106–122. DOI: 10.1016/j.matpur.2018.04.010, https://doi.org/10.1016/j.matpur.2018.04.010 (preprint arXiv:1801.10330, https://arxiv.org/abs/1801.10330)
6. Claude Le Bris, 'Defects in homogenization theory', Séminaire Laurent Schwartz — EDP et applications (2022–2023), exposé no II, 1–17. DOI: 10.5802/slsedp.157, https://doi.org/10.5802/slsedp.157 (full text at https://proceedings.centre-mersenne.org/item/10.5802/slsedp.157.pdf)
7. Rémi Goudey, 'A periodic homogenization problem with defects rare at infinity', Netw. Heterog. Media 17 (2022) 547–592. DOI: 10.3934/nhm.2022014, https://doi.org/10.3934/nhm.2022014 (preprint arXiv:2109.05506, https://arxiv.org/abs/2109.05506)
8. Rémi Goudey, 'Elliptic homogenization with almost translation-invariant coefficients', Asymptot. Anal. 132 (2022) 175–216. DOI: 10.3233/ASY-221789, https://doi.org/10.3233/ASY-221789 (preprint arXiv:2202.07492, https://arxiv.org/abs/2202.07492)
9. Xavier Blanc, Marc Josien, Claude Le Bris, 'Precised approximations in elliptic homogenization beyond the periodic setting', Asymptot. Anal. 116 (2020) 93–137. DOI: 10.3233/ASY-191537, https://doi.org/10.3233/ASY-191537 (preprint arXiv:1812.07220, https://arxiv.org/abs/1812.07220)
10. Sylvain Wolf, 'Homogenization of the p-Laplace equation in a periodic setting with a local defect', Nonlinear Anal. 228 (2023) 113182. DOI: 10.1016/j.na.2022.113182, https://doi.org/10.1016/j.na.2022.113182 (preprint arXiv:2206.03071, https://arxiv.org/abs/2206.03071)
11. Yves Achdou, Claude Le Bris, 'Homogenization of some periodic Hamilton-Jacobi equations with defects', Commun. Partial Differ. Equ. 48 (2023) 944–986. DOI: 10.1080/03605302.2023.2238953, https://doi.org/10.1080/03605302.2023.2238953 (preprint arXiv:2211.16157, https://arxiv.org/abs/2211.16157)
12. Yves Achdou, Claude Le Bris, 'Homogenization of Hamilton-Jacobi equations with defects leading to stratified problems', NoDEA Nonlinear Differential Equations Appl. 32 (2025), article 119. DOI: 10.1007/s00030-025-01105-y, https://doi.org/10.1007/s00030-025-01105-y (preprint arXiv:2412.07280, https://arxiv.org/abs/2412.07280)
13. Pierre Cardaliaguet, Claude Le Bris, Panagiotis E. Souganidis, 'Perturbation problems in homogenization of Hamilton-Jacobi equations', J. Math. Pures Appl. (9) 117 (2018) 221–262. DOI: 10.1016/j.matpur.2018.03.005, https://doi.org/10.1016/j.matpur.2018.03.005
14. Lutz Recke, 'A common approach to singular perturbation and homogenization III: Nonlinear periodic homogenization with localized defects', preprint (2025). arXiv:2502.13169, https://arxiv.org/abs/2502.13169
15. Jizu Huang, Yong Ma, 'A two-dimensional structural local-defect theory for scalar non-divergence advection-diffusion homogenization', preprint (2026). arXiv:2607.02979, https://arxiv.org/abs/2607.02979
16. Achille Landri Pokam Kakeu, Mapundi Kondwani Banda, 'Existence and uniqueness of correctors for nonlinear homogenization in uniformly local Sobolev spaces', The Journal of Analysis (2026). DOI: 10.1007/s41478-026-01153-2, https://doi.org/10.1007/s41478-026-01153-2
17. Mathias Schäffner, Ben Schweizer, Yohanes Tjandrawidjaja, 'Domain truncation methods for the wave equation in a homogenization limit', Appl. Anal. 101 (2022) 4149–4170. DOI: 10.1080/00036811.2022.2054416, https://doi.org/10.1080/00036811.2022.2054416
18. Marco Avellaneda, Fang-Hua Lin, 'Compactness methods in the theory of homogenization', Commun. Pure Appl. Math. 40 (1987) 803–847. DOI: 10.1002/cpa.3160400607, https://doi.org/10.1002/cpa.3160400607
19. V. V. Jikov, S. M. Kozlov, O. A. Oleinik, 'Homogenization of Differential Operators and Integral Functionals', Springer, Berlin, 1994. DOI: 10.1007/978-3-642-84659-5, https://doi.org/10.1007/978-3-642-84659-5