Quantitative error analysis for Crandall–Lions monotone approximations of mean curvature motion: sharp rates, practical parameter couplings, and the nonlocal equation

Crandall and Lions (Numer. Math. 75 (1996) 17–41) constructed the first provably convergent explicit finite-difference schemes for the level-set mean-curvature equation and for the degenerate quasilinear parabolic class $u_t - \operatorname{trace}(\Theta(x,Du)\Theta(x,Du)^{\mathrm T}D^2u) = 0$. Their convergence theorem — recorded by Deckelnick (2000) as the first convergence proof for a scheme approximating the level-set MCM equation — came without quantitative analysis: no convergence rate for the regularized solutions, no a priori error bound linking the discretization parameters to the accuracy, and extremely restrictive time-step conditions. The same paper also analyzed a nonlocal problem related to the schemes and presented a further, very different sort of approximation. Later work resolved the first-order version of the rates question: Deckelnick (2000) proved $\|u - u^\epsilon\|_\infty \le C\epsilon^\alpha$ for every $\alpha < 1/2$ and an $L^\infty$ error bound for the Crandall–Lions scheme itself under an explicit but very restrictive parameter coupling ($\Delta t \sim \epsilon^{17/2}$); Mitake (2011) simplified the argument and extended the rate to power-curvature equations; Forcadel (2008) obtained error estimates for a different anisotropic scheme with a weaker logarithmic-type rate; Kurkinen–Liu (2026) recovered the same rate inside a general quantitative-stability framework while explicitly not addressing optimality. The open directions that remain unsettled are: (i) sharpness of the $1/2$-exponent barrier for the regularization (is $O(\epsilon^{1/2})$ attained, and is $1/2$ optimal?); (ii) a priori error bounds under practical time-step restrictions and for bounded uniformly continuous data; (iii) quantitative modulus-of-continuity estimates for the Crandall–Lions nonlocal equation; (iv) rates and constructive parameter bounds for the $(x,p)$-dependent and fully nonlinear (Isaacs) class in space dimension at least two.

Background

Motion by mean curvature is the geometric evolution law in which each point of a hypersurface moves along its normal with speed equal to the mean curvature at that point. In the level-set approach the evolving surface is the zero level set of a function $u$ solving the mean-curvature motion (MCM) equation

$$u_t - \operatorname{trace}\Big(\Big(I - \frac{Du \otimes Du}{|Du|^2}\Big)D^2u\Big) = 0,$$

where $I$ is the identity matrix and $D^2u$ the Hessian. The matrix $I - Du \otimes Du/|Du|^2$ projects onto the tangent plane of the level sets, so the equation is degenerate (diffusion is suppressed in the normal direction) and singular wherever $Du = 0$. Well-posedness for bounded uniformly continuous (BUC) data holds in the viscosity sense — a stability-based weak-solution notion defined by one-sided inequalities against smooth test functions — as established by Chen–Giga–Goto and Evans–Spruck (1991).

Crandall and Lions (1996) studied the class (PE) $u_t - \operatorname{trace}(\Theta(x,Du)\Theta(x,Du)^{\mathrm T}D^2u) = 0$ with $\Theta$ an $N \times M$ matrix-valued function (MCM corresponds to $\Theta\Theta^{\mathrm T} = I - p \otimes p/|p|^2$), for which $\Theta\Theta^{\mathrm T}$ is merely nonnegative: the equation is degenerate parabolic. They constructed a family of explicit finite-difference schemes $W_{\Delta t} = \tilde U_{\Delta t} + V_{\Delta t}$: directional second differences of width $h$ are taken along the columns of a regularized coefficient matrix $\Theta_\epsilon(p) = I - p \otimes p/(|p|^2 + \epsilon)$ (an Evans–Spruck-type regularization of the singularity at $p = 0$; the equivalent Evans–Spruck/Deckelnick form replaces $|p|^2$ by $|p|^2 + \epsilon^2$, the same cutoff under $\epsilon \mapsto \epsilon^2$, and the convergence-rate literature below states all exponents in that $\epsilon^2$-parameterization); values at points off the grid of mesh size $\rho$ are supplied by multilinear interpolation; and the monotonicity lost to the interpolation is restored by an artificial-viscosity term $V_{\Delta t}$ (coordinate-direction second differences with coefficient $\rho K/h$). The scheme is monotone (order-preserving), consistent, and stable provided the parameters satisfy CFL-type relations (CFL: Courant–Friedrichs–Lewy restrictions bounding the time step in terms of the spatial mesh) such as $2N\Delta t(1/h^2 + K/(h\rho)) \le 1$ together with $K$ of order $\epsilon^{-1/2}$, $\rho/h^2 \to 0$ and $\rho K/h \to 0$ — restrictions under which the admissible time step $\Delta t$ is forced to be extremely small. By the Barles–Souganidis framework (monotone + consistent + stable one-step schemes converge to the viscosity solution), they proved convergence for BUC initial data — the first convergence proof for a scheme approximating the level-set MCM equation — but provided no convergence rate, no error bound, and no quantitative law for choosing the five parameters $(\Delta t, h, \rho, K, \epsilon)$ to reach a given accuracy. The same paper introduced the nonlocal stationary equation

$$u(x) - \sum_k \lambda_k\, u\big(x + a_k(x, Du(x))\big) = f(x)$$

arising from implicit-in-time discretizations, for which they proved existence and uniqueness by a contraction argument, and asserted that the scheme extends to fully nonlinear equations of Isaacs type (the sup–inf form arising in two-player differential games) $\sup_a \inf_b \operatorname{trace}(\Theta^{a,b}(x,Du)\Theta^{a,b}(x,Du)^{\mathrm T}D^2u) = 0$ under uniform hypotheses.

Two quantitative threads were developed afterwards. First, for the regularization $u^\epsilon$ of (MC) (the solution with $\Theta$ replaced by $\Theta_\epsilon$), Deckelnick (2000) proved $\|u - u^\epsilon\|_\infty \le C\epsilon^\alpha$ for every $\alpha \in (0,1/2)$ (smooth, compactly constant data), using curvature bounds for evolving graphs to control higher derivatives of $u^\epsilon$; and, for the Crandall–Lions scheme itself, the discrete error bound $\sup_{x \in \mathscr G_\rho, l}|U^l(x) - u(x,t_l)| \le C\epsilon^\alpha$ under the explicit coupling $h = \epsilon^{5/2}$, $\rho = \epsilon^{11/2}$, $\Delta t = c_1\epsilon^{17/2}$, $K = c_2\epsilon^{-1/2}$. Mitake (2011) gave a simpler proof with an explicit constant for Lipschitz data and extended the rate analysis to power-curvature equations $V = \kappa^\alpha$ (normal velocity equal to a power of the mean curvature $\kappa$), including the affine-curvature equation ($\alpha = 1/3$). Second, Forcadel (2008) proved error estimates for a new scheme for anisotropic MCM derived from dislocation dynamics, with a logarithmic-type rate; Carlini–Falcone–Ferretti (2010) analyzed semi-Lagrangian large-time-step schemes (schemes whose update interpolates the previous iterate at finitely many characteristic foot-points) that avoid the parabolic stability condition $\Delta t/\Delta x^2 \le C$, proved their convergence, and explicitly left a priori error estimates open; Kurkinen–Liu (2026) placed the $O(\epsilon^\nu)$, $\nu < 1/2$ regularization rate within a general quantitative-stability theory for quasilinear parabolic equations with singular coefficients, extending it to Hölder solutions, while stating that they do not seek optimality of the rates. What remains open is the sharpness and the practical reach of this quantitative theory; the precise open content is stated in the problem statement.

Problem Statement

The Crandall–Lions program established convergence, but not quantitative error control, for monotone approximations of the level-set mean-curvature equation

$$u_t - \operatorname{trace}\Big(\Big(I - \frac{Du \otimes Du}{|Du|^2}\Big)D^2u\Big) = 0,\qquad u(\cdot,0) = u^0 \in \mathrm{BUC}(\mathbb R^N),$$

where $\mathrm{BUC}$ denotes the bounded uniformly continuous functions; the equation is called (MC) below, and of the wider class (PE) $u_t - \operatorname{trace}(\Theta(x,Du)\Theta(x,Du)^{\mathrm T}D^2u) = 0$ with $\Theta$ matrix-valued and $\Theta\Theta^{\mathrm T}$ merely nonnegative (degenerate). The documented quantitative state of the art is: $\|u - u^\epsilon\|_\infty \le C\epsilon^\alpha$ for every $\alpha \in (0,1/2)$ for the Evans–Spruck/Crandall–Lions regularization — $u^\epsilon$ solves (MC) with $I - Du \otimes Du/|Du|^2$ replaced by $I - Du \otimes Du/(|Du|^2 + \epsilon^2)$ (the Crandall–Lions paper writes the cutoff as $|Du|^2 + \epsilon$, the same family under $\epsilon \mapsto \epsilon^2$; all exponents in this problem are stated in the $\epsilon^2$-parameterization) — for smooth or Lipschitz data (Deckelnick 2000, Mitake 2011, Kurkinen–Liu 2026); and the discrete error bound $\sup|U^l - u(\cdot,t_l)| \le C\epsilon^\alpha$ for the Crandall–Lions scheme under the coupling $h = \epsilon^{5/2}$, $\rho = \epsilon^{11/2}$, $\Delta t = c_1\epsilon^{17/2}$, $K = c_2\epsilon^{-1/2}$ (Deckelnick 2000, $C^4$ data; $h$ is the directional-difference width, $\rho$ the interpolation mesh size, $K$ the artificial-viscosity coefficient, $\Delta t$ the time step, and $\epsilon$ the regularization parameter). The open problem is to complete the quantitative theory; the following linked components remain open against that baseline:

(1) Sharpness of the regularization rate. Determine whether the exponent $1/2$ is attained — does there exist $C$ with $\sup_{\mathbb R^N \times [0,T]} |u - u^\epsilon| \le C\epsilon^{1/2}$ for a natural class of data (e.g. Lipschitz or BUC)? — and whether $1/2$ is optimal: either construct initial data for which the error is not $o(\epsilon^{1/2})$ — that is, $\sup |u - u^\epsilon| \ge c\,\epsilon^{1/2}$ for some $c > 0$, along a sequence $\epsilon \downarrow 0$ if necessary — or prove that no lower bound at the exponent $1/2$ holds, in the sense that $\sup |u - u^\epsilon| = o(\epsilon^{1/2})$ for the data class in question. (A lower bound $\sup |u - u^\epsilon| \ge c\,\epsilon^\beta$ with $\beta > 1/2$ would not by itself establish optimality, since it is consistent with an $O(\epsilon^{1/2})$ upper bound.) No result in the literature reaches the exponent $1/2$, and none addresses optimality.

(2) A priori error bounds under practical couplings. Prove $L^\infty$ error estimates for monotone (or generalized-monotone) approximations of (MC) — the Crandall–Lions scheme itself, or explicit large-time-step, semi-Lagrangian, or implicit variants of the Crandall–Lions construction — under parameter couplings strictly weaker than Deckelnick's $\Delta t = c_1\epsilon^{17/2}$, in the sense that the admissible time step degrades more slowly than $\epsilon^{17/2}$ as $\epsilon \downarrow 0$, and ideally for data in the BUC class of the convergence theorem rather than only $C^4$/Lipschitz data. Carlini–Falcone–Ferretti (2010) explicitly leave the a priori error estimate for their large-time-step scheme open.

(3) Quantitative regularity for the nonlocal equation. For the Crandall–Lions nonlocal equation

$$u(x) - \sum_k \lambda_k\, u\big(x + a_k(x, Du(x))\big) = f(x),\qquad \sum_k \lambda_k < 1,$$

with nonnegative weights $\lambda_k$ and shift maps $a_k(x,p)$, arising from implicit-in-time discretizations of the regularized equations, establish a priori modulus-of-continuity estimates for its viscosity solutions, beyond the contraction-based existence and uniqueness theory, of the kind needed to make implicit or semi-Lagrangian discretizations of (MC) quantitatively analyzable.

(4) Constructive quantitative bounds for the general class. For the $(x,p)$-dependent quasilinear case $\Theta(x,p)$ and the fully nonlinear Isaacs case (sup–inf over two-player differential-game controls) $\sup_a \inf_b \operatorname{trace}(\Theta^{a,b}(x,Du)\Theta^{a,b}(x,Du)^{\mathrm T}D^2u) = 0$, construct monotone approximations with proven convergence rates and explicit, constructively checkable parameter relations in space dimension $N \ge 2$ (rates for non-convex degenerate second-order equations are known only in one space dimension, and in arbitrary dimension only for the convex Bellman case).

The common objective is to bring the Crandall–Lions monotone approximations of curvature-type degenerate parabolic equations from mere convergence to an a priori error theory: explicit rates, optimal parameter-scaling laws, and quantitative stability constants. Components (1)–(4) are each independently open: none is a prerequisite for another. Within component (1), the two questions are whether the exponent $1/2$ is attained and whether it is optimal.

References

1. Michael G. Crandall and Pierre-Louis Lions, Convergent difference schemes for nonlinear parabolic equations and mean curvature motion, Numerische Mathematik 75(1):17–41, 1996. DOI: 10.1007/s002110050228. https://doi.org/10.1007/s002110050228
2. L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, Journal of Differential Geometry 33, 1991. DOI: 10.4310/jdg/1214446559. https://doi.org/10.4310/jdg/1214446559 (The regularization and its convergence to the MCM solution.)
3. Yun-Gang Chen, Yoshikazu Giga, and Shun-ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Journal of Differential Geometry 33, 1991. DOI: 10.4310/jdg/1214446564. https://doi.org/10.4310/jdg/1214446564
4. G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis 4, 1991. DOI: 10.3233/ASY-1991-4305. https://doi.org/10.3233/ASY-1991-4305 (Convergence framework for these monotone approximation schemes.)
5. Klaus Deckelnick, Error bounds for a difference scheme approximating viscosity solutions of mean curvature flow, Interfaces and Free Boundaries 2(2):117–142, 2000. DOI: 10.4171/IFB/15. https://doi.org/10.4171/IFB/15
6. Nicolas Forcadel, An error estimate for a new scheme for mean curvature motion, SIAM Journal on Numerical Analysis 46(5):2715–2741, 2008. DOI: 10.1137/060678282. https://doi.org/10.1137/060678282
7. Hiroyoshi Mitake, On convergence rates for solutions of approximate mean curvature equations, Proceedings of the American Mathematical Society 139(10):3691–3696, 2011. DOI: 10.1090/S0002-9939-2011-11002-5. https://doi.org/10.1090/S0002-9939-2011-11002-5
8. Adam M. Oberman, A convergent monotone difference scheme for motion of level sets by mean curvature, Numerische Mathematik 99:365–379, 2004. DOI: 10.1007/s00211-004-0566-1. https://doi.org/10.1007/s00211-004-0566-1
9. Elisabetta Carlini, Maurizio Falcone, and Roberto Ferretti, Convergence of a large time-step scheme for mean curvature motion, Interfaces and Free Boundaries 12(4):409–441, 2010. DOI: 10.4171/IFB/240. https://doi.org/10.4171/IFB/240
10. Robert Kohn and Sylvia Serfaty, A deterministic-control-based approach to motion by curvature, Communications on Pure and Applied Mathematics 59, 2006. DOI: 10.1002/cpa.20101. https://doi.org/10.1002/cpa.20101 (Min–max game approximation of motion by curvature.)
11. Espen R. Jakobsen, On error bounds for approximation schemes for non-convex degenerate elliptic equations, BIT Numerical Mathematics 44, 2004. DOI: 10.1023/B:BITN.0000039390.33444.F2. https://doi.org/10.1023/B:BITN.0000039390.33444.F2
12. Tapio Kurkinen and Qing Liu, Quantitative stability for quasilinear parabolic equations, preprint, 2026. arXiv:2602.12657. https://arxiv.org/abs/2602.12657
13. Axel Kröner and Heiko Kröner, A priori error estimates for finite element approximations of regularized level set flows in higher norms, Applied Numerical Mathematics 171:307–328, 2022. DOI: 10.1016/j.apnum.2021.09.008. https://doi.org/10.1016/j.apnum.2021.09.008