Is the coercivity-type condition (1.13) of Lions–Souganidis (2005) necessary for homogenization of almost periodic degenerate quasilinear equations when $N>1$?

Lions and Souganidis (Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005) 667–677) proved homogenization for the degenerate quasilinear equation $-\varepsilon\,\mathrm{tr}\,A(Du^{\varepsilon},x,\varepsilon^{-1}x)D^{2}u^{\varepsilon}+H(Du^{\varepsilon},u^{\varepsilon},x,\varepsilon^{-1}x)=0$ in periodic and almost periodic media under a coercivity-type growth condition coupling the Hamiltonian $H$ and the diffusion matrix $\Sigma$ (their condition (1.13)), and explicitly asked whether, in space dimension $N>1$, this condition is actually necessary for the homogenization of almost periodic quasi-linear equations — they show it is not necessary when $N=1$. The geometric sibling (necessity of the analogous condition (1.20) for forced mean curvature flow) was resolved by Caffarelli and Monneau: in $N\ge3$ periodic media a condition of that type is genuinely necessary, while in $N=2$ it essentially is not. The random-media issue was substantially advanced by Lions–Souganidis, Kosygina–Rezakhanlou–Varadhan and Armstrong–Cardaliaguet. The almost periodic quasilinear necessity question itself, however, remains open: no counterexample and no homogenization theorem under weaker assumptions is known, and the companion counterexample announced by the authors as their reference [28] ('(1.13) is indeed necessary to have uniform Lipschitz estimates for $N>1$') was never published. The open problem is to prove or disprove the necessity of (1.13) for homogenization of almost periodic quasilinear equations when $N>1$.

Background

Homogenization studies the limiting behavior of solutions of PDE with rapidly oscillating coefficients. In the setting of the source paper (Lions–Souganidis 2005) one considers, for $\varepsilon>0$, the equation $F(\varepsilon D^{2}u^{\varepsilon},Du^{\varepsilon},u^{\varepsilon},x,\varepsilon^{-1}x)=0\ \text{in}\ \mathbb{R}^{N},$ where the dependence on $\varepsilon^{-1}x$ models a medium oscillating at scale $\varepsilon$ (eq. (1.1) of the paper), and asks whether $u^{\varepsilon}$ converges locally uniformly, as $\varepsilon\to0$, to the solution of an effective equation with coefficients averaged over the medium. Solutions are understood in the viscosity sense — the standard weak-solution class for degenerate elliptic equations, introduced and surveyed by Crandall, Ishii and Lions. The paper focuses on the quasilinear degenerate class $-\varepsilon\,\mathrm{tr}\,A(Du^{\varepsilon},x,\varepsilon^{-1}x)\,D^{2}u^{\varepsilon}+H(Du^{\varepsilon},u^{\varepsilon},x,\varepsilon^{-1}x)=0,$ (eq. (1.7)) where $A=\Sigma\Sigma^{T}$ with $\Sigma$ an $N\times M$ matrix-valued function, locally Lipschitz in $(p,x,y)$ and Lipschitz in $(x,y)$ uniformly for $p$ in bounded sets (the paper's assumption (1.12)); $A$ is positive semidefinite but may be rank-deficient or vanish — degenerate ellipticity, so no uniform ellipticity and no Krylov–Safonov-type regularity theory is available — and $H$ is uniformly proper in $u$ (monotone in the $u$-argument, which yields comparison principles). Two media classes are treated: periodic, where the coefficients descend to a compact torus and the cell problem is solved on that torus, and almost periodic (in Bohr's sense: uniform limits of trigonometric polynomials, equivalently families of translates with a common compact approximation set — a spatial recurrence condition strictly weaker than periodicity). In the almost periodic case there is no compact torus; the authors approximate correctors by truncated problems on large balls, and the crux is a uniform Lipschitz bound on these approximate correctors, obtained by a Bernstein-type method (a barrier/test-function technique for gradient estimates; the classical version needs $C^{2}$ correctors, which for nonconvex $F$ is not available, so a weak Bernstein argument is used). Their key sufficient condition, (1.13), is a coercivity condition on the joint growth of $H$ and $\Sigma$ in $p$: there exists $\theta\in(0,1)$ such that $\lim_{|p|\to\infty}|p|^{-2}\Big[\theta(1-\theta)H^{2}+\theta\|\Sigma\|^{2}\,D_{y}H\cdot p\Big]>\|D_{y}\Sigma\|^{2}\,\|\Sigma\|^{2},$ where $H=H(p,r,x,y)$, $\Sigma=\Sigma(p,x,y)$, $\|\Sigma\|^{2}=\mathrm{tr}\,A$, and repeated indices are summed. Heuristically it requires the driving Hamiltonian to grow fast enough relative to the oscillation of the diffusion matrix in the fast variable $y$; it holds automatically when $H$ grows superlinearly in $p$. The paper proves homogenization under (1.13) and records the one-dimensional counterpoint: for $-a(y,v)v''+b(y)|v'|=0$ on $\mathbb{R}$ with $b>0$, the bound $\inf b>0$ alone already yields a Lipschitz estimate, so (1.13) is not necessary when $N=1$.

The same paper treats the geometric (level-set) equation for forced mean curvature flow of interfaces, $u^{\varepsilon}_{t}-\varepsilon\delta\,\mathrm{tr}\big[(I-\hat{Du}^{\varepsilon}\otimes\hat{Du}^{\varepsilon})D^{2}u^{\varepsilon}\big]+a(\varepsilon^{-1}x)H(Du^{\varepsilon})=0$ (eq. (1.17)), where $\hat{p}=p/|p|$, $\delta>0$ is a surface-tension coefficient, $a$ is an almost periodic velocity with $\inf a>0$, and $H(p)=|p|v(\hat{p})$ with $v>0$; its sufficient condition (1.20) reads $\inf_{y}\big[a^{2}(y)v-(N-1)\delta|D_{y}a(y)|\big]>0$. The authors conjectured that failure of such conditions is related to pinning — interfaces in heterogeneous media remaining stuck despite positive average driving force, a phenomenon documented for pulsating waves (periodic traveling fronts of geometric flows) by Dirr, Karali and Yip, analyzed through a homogenize / trap / perpetually oscillate trichotomy by Cardaliaguet, Lions and Souganidis and by Cesaroni and Novaga, and surveyed by Caffarelli. That conjecture was confirmed: Caffarelli and Monneau (2014) constructed, in $N\ge3$, a smooth periodic velocity with positive infimum for which homogenization of forced mean curvature flow fails — so in the geometric class a (1.20)-type condition is genuinely necessary in $N\ge3$ — while in $N=2$ they proved homogenization under mild additional structure assumptions. In the random direction, Lions and Souganidis (CPDE 2005) and Kosygina, Rezakhanlou and Varadhan proved stochastic homogenization of uniformly parabolic Hamilton–Jacobi–Bellman equations in stationary ergodic media (random media whose law is translation-invariant and ergodic), Armstrong and Souganidis extended this to unbounded and degenerate-elliptic Bellman equations, Armstrong and Cardaliaguet (2018) proved the first stochastic homogenization results for the $\varepsilon$-scaled quasilinear and positively homogeneous class under finite range of dependence (a quantitative mixing assumption), and Ziliotto and Feldman–Fermanian–Ziliotto exhibited stationary ergodic media and nonconvex Hamiltonians (first-order and viscous, respectively) for which homogenization fails. Jang (2025) showed that in periodic media geometric and positively homogeneous equations homogenize under cell-problem solvability alone, with no coercivity condition. Against this backdrop, the smooth almost periodic quasilinear class is the least understood: whether (1.13) is necessary there, in either the homogenization or the Lipschitz-estimate sense, is untouched by all of these developments.

Problem Statement

Prove or refute, for every $N>1$, the necessity of condition (1.13) for the homogenization of almost periodic quasi-linear degenerate elliptic equations, i.e. whether homogenization in this class implies (1.13). This is the question left open by Lions–Souganidis (2005), stated in their own words: 'We do not know, however, whether, when $N>1$, this condition is actually necessary for the homogenization of almost periodic quasi-linear equations.'

Setting. For $\varepsilon>0$ consider $-\varepsilon\,\mathrm{tr}\,A(Du^{\varepsilon},x,\varepsilon^{-1}x)\,D^{2}u^{\varepsilon}+H(Du^{\varepsilon},u^{\varepsilon},x,\varepsilon^{-1}x)=0\ \text{in}\ \mathbb{R}^{N},$ in the class of the source paper's Theorem 1.1, i.e. satisfying its assumptions (1.8)–(1.12): $H=H(p,r,x,y)$ (with $r$ the unknown's value and $p$ its gradient) and $A=A(p,x,y)$ are locally Lipschitz in their arguments and Lipschitz in $(r,x,y)$ uniformly for $p$ in bounded sets; $|D_{x}H|+|D_{y}H|\le C(1+|H|)$ for some constant $C$; $H$ is uniformly proper in $r$, i.e. there exists $\gamma>0$ such that $H(p,r,x,y)-H(p,s,x,y)\ge\gamma(r-s)$ whenever $r\ge s$; $H$ and $A$ are almost periodic in the fast variable $y$ (Bohr almost periodic), locally uniformly in the other variables; and $A=\Sigma\Sigma^{T}$ for an $N\times M$ matrix-valued $\Sigma(p,x,y)$ in the same regularity class, so $A$ is positive semidefinite but may be rank-deficient or vanish (degenerate ellipticity). Condition (1.13) is the object of the necessity question and, together with (1.9), implies the coercivity (1.14) that $H(p,r,x,y)\to\infty$ as $|p|\to\infty$ uniformly in $(r,x,y)$; since the necessity question drops (1.13), condition (1.14) is imposed here as a standing hypothesis of the class, while $A$ remains allowed to be rank-deficient or vanish, and the data are further required to be non-degenerate in that the $\varepsilon$-equations admit a bounded uniformly continuous viscosity solution that is not constant, so that homogenization is not vacuous. Say that homogenization holds for the data $(\Sigma,H)$ if there exists an effective (non-oscillatory, $y$- and $\varepsilon$-independent) equation $\bar F(D\bar u,\bar u,x)=0$ with $\bar F$ in the class of the source paper's effective operator (its Theorem 1.1), i.e. $\bar F$ is locally Lipschitz, uniformly proper in the $\bar u$-argument (as in assumption (1.10): there exists $\gamma>0$ such that $\bar F(p,r,x)-\bar F(p,s,x)\ge\gamma(r-s)$ whenever $r\ge s$) and coercive in the gradient argument (as in (1.14): $\bar F(p,r,x)\to\infty$ as $|p|\to\infty$, uniformly in $r$ and $x$), such that the effective equation admits a bounded uniformly continuous viscosity solution $\bar u$ and, whenever the $\varepsilon$-equations admit bounded uniformly continuous viscosity solutions $u^{\varepsilon}$, one has $u^{\varepsilon}\to\bar u$ locally uniformly as $\varepsilon\to0$. The sufficient condition (1.13) of the source paper is: there exists $\theta\in(0,1)$ such that $\lim_{|p|\to\infty}|p|^{-2}\Big[\theta(1-\theta)H^{2}+\theta\|\Sigma\|^{2}\,D_{y}H\cdot p\Big]>\|D_{y}\Sigma\|^{2}\,\|\Sigma\|^{2},$ with $\|\Sigma\|^{2}=\mathrm{tr}\,A$ and summation over repeated indices (in the general-$F$ case the corresponding condition is (1.24) of the same paper).

The question has two admissible resolutions, in either direction and without extra restrictions on the data beyond the class above; they are the statement of the question and its exact logical negation, with the dimension quantifier fixed in each:

1. Necessity (proof of the implication): prove that homogenization in the class implies (1.13) — equivalently, that for every $N>1$ no data $(\Sigma,H)$ of the class violating (1.13) homogenize, i.e. each such datum either admits a family of $\varepsilon$-solutions with no locally uniform effective limit, or exhibits a failure of the uniform Lipschitz a priori estimate in a form that precludes any effective equation. A single datum violating (1.13) for which homogenization fails is consistent with both answers and by itself settles neither side.
2. Non-necessity (refutation of the implication): refute the implication by exhibiting, for some $N>1$, explicit data $(\Sigma,H)$ in the class that satisfy assumptions strictly weaker than (1.13), violate (1.13), and homogenize. A weaker, publicly unproven milestone recorded in the source paper is the Lipschitz-estimate form of the necessity question: when $N>1$, is (1.13) necessary for uniform Lipschitz estimates for the (approximate) correctors? The source attributes an affirmative answer to a companion paper 'in preparation' (its reference [28]) which was never published; a proof or refutation of that claim would be the first public evidence on the necessity side. Resolving the Lipschitz-estimate milestone alone does not resolve the task; a resolution must address homogenization, as in item 1 or item 2. The one-dimensional case is settled — (1.13) is not necessary when $N=1$ — so the question is genuinely about $N>1$.

References

1. P.-L. Lions and P. E. Souganidis, 'Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications', Ann. Inst. H. Poincaré Anal. Non Linéaire 22(5) (2005) 667–677. DOI: 10.1016/j.anihpc.2004.10.009. URL: https://doi.org/10.1016/j.anihpc.2004.10.009
2. L. A. Caffarelli and R. Monneau, 'Counter-Example in Three Dimension and Homogenization of Geometric Motions in Two Dimension', Arch. Ration. Mech. Anal. 212 (2014) 503–574. DOI: 10.1007/s00205-013-0712-y. URL: https://doi.org/10.1007/s00205-013-0712-y
3. S. N. Armstrong and P. Cardaliaguet, 'Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions', J. Eur. Math. Soc. 20 (2018) 797–864. DOI: 10.4171/jems/777. URL: https://doi.org/10.4171/jems/777
4. J. Jang, 'Periodic homogenization of geometric equations without perturbed correctors', Math. Ann. 391 (2025) 3143–3180. DOI: 10.1007/s00208-024-02998-3. URL: https://doi.org/10.1007/s00208-024-02998-3
5. A. Davini, 'Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d', Calc. Var. Partial Differential Equations 64 (2025), article 38. DOI: 10.1007/s00526-024-02870-x. URL: https://doi.org/10.1007/s00526-024-02870-x
6. B. Ziliotto, 'Stochastic Homogenization of Nonconvex Hamilton–Jacobi Equations: A Counterexample', Comm. Pure Appl. Math. 70 (2017) 1798–1809. DOI: 10.1002/cpa.21674. URL: https://doi.org/10.1002/cpa.21674
7. W. M. Feldman, J.-B. Fermanian and B. Ziliotto, 'An example of failure of stochastic homogenization for viscous Hamilton–Jacobi equations without convexity', J. Differential Equations 280 (2021) 464–476. DOI: 10.1016/j.jde.2021.01.009. URL: https://doi.org/10.1016/j.jde.2021.01.009
8. W. M. Feldman, 'Mean curvature flow with positive random forcing in 2-d', preprint (2019). arXiv:1911.00488. URL: https://arxiv.org/abs/1911.00488
9. P. S. Morfe, 'On the homogenization of second order level set PDE in periodic media', preprint (2020). arXiv:2011.15062. URL: https://arxiv.org/abs/2011.15062
10. H. Mitake, C. Mooney, H. V. Tran, J. Xin and Y. Yu, 'Bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows', Math. Ann. 391 (2025) 3077–3111. DOI: 10.1007/s00208-024-02983-w. URL: https://doi.org/10.1007/s00208-024-02983-w
11. P. Cardaliaguet, P.-L. Lions and P. E. Souganidis, 'A discussion about the homogenization of moving interfaces', J. Math. Pures Appl. 91 (2009) 339–363. DOI: 10.1016/j.matpur.2009.01.014. URL: https://doi.org/10.1016/j.matpur.2009.01.014
12. L. A. Caffarelli, 'The homogenization of surfaces and boundaries', Bull. Braz. Math. Soc. (N.S.) 44 (2013) 755–775. DOI: 10.1007/s00574-013-0033-7. URL: https://doi.org/10.1007/s00574-013-0033-7
13. E. Kosygina, F. Rezakhanlou and S. R. S. Varadhan, 'Stochastic homogenization of Hamilton–Jacobi–Bellman equations', Comm. Pure Appl. Math. 59 (2006) 1489–1521. DOI: 10.1002/cpa.20137. URL: https://doi.org/10.1002/cpa.20137
14. P.-L. Lions and P. E. Souganidis, 'Homogenization of "viscous" Hamilton–Jacobi equations in stationary ergodic media', Comm. Partial Differential Equations 30 (2005) 335–375. DOI: 10.1081/PDE-200050077. URL: https://doi.org/10.1081/PDE-200050077
15. S. N. Armstrong and P. E. Souganidis, 'Stochastic homogenization of Hamilton–Jacobi and degenerate Bellman equations in unbounded environments', J. Math. Pures Appl. 97 (2012) 460–504. DOI: 10.1016/j.matpur.2011.09.009. URL: https://doi.org/10.1016/j.matpur.2011.09.009
16. N. Dirr, G. Karali and N. K. Yip, 'Pulsating wave for mean curvature flow in inhomogeneous medium', European J. Appl. Math. 19 (2008) 661–699. DOI: 10.1017/S095679250800764X. URL: https://doi.org/10.1017/S095679250800764X
17. A. Cesaroni and M. Novaga, 'Long-Time Behavior of the Mean Curvature Flow with Periodic Forcing', Comm. Partial Differential Equations 38 (2013) 780–801. DOI: 10.1080/03605302.2013.771508. URL: https://doi.org/10.1080/03605302.2013.771508
18. M. G. Crandall, H. Ishii and P.-L. Lions, 'User's guide to viscosity solutions of second order partial differential equations', Bull. Amer. Math. Soc. 27 (1992) 1–67. DOI: 10.1090/S0273-0979-1992-00266-5. URL: https://doi.org/10.1090/S0273-0979-1992-00266-5