Stern's Level Set Method for Mean Curvature Flow

• Stern's Level Set Method is a level-set formulation for mean curvature flow that uses an obstacle constraint to enforce prescribed fixed boundaries.
• It employs viscosity solution techniques to guarantee existence, uniqueness, and Lipschitz regularity of the evolving hypersurface.
• The method exhibits sharp consistency with classical Dirichlet-boundary flows and the Sternberg–Ziemer construction in mean-convex domains.

The Stern level-set method, as developed in "A level-set method for a mean curvature flow with a prescribed boundary" by Bian, Giga, and Mitake, provides a global-in-time level-set approach for mean curvature flow with a prescribed geometric boundary by formulating the boundary as an obstacle constraint. This method achieves global solvability for general initial hypersurfaces and ensures sharp consistency with both classical Dirichlet-boundary mean curvature flow and the Sternberg–Ziemer (1994) Dirichlet-flow in strictly mean-convex domains (Bian et al., 2023).

1. PDE Formulation and Obstacle Boundary Interpretation

The method describes the evolution of a family of hypersurfaces $\{\Gamma_t\}\subset \mathbb{R}^n$ governed by the mean curvature flow law

$$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$

where $V$ is normal velocity and $H = \operatorname{div}\nu$ the sum of principal curvatures for unit normal $\nu$. The level-set approach introduces a scalar function $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$ so that

$\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$

with $u < 0$ and $u > 0$ on either side, respectively. The governing degenerate parabolic PDE is

$$u_t - |\nabla u|\operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) = 0, \quad \text{in } \mathbb{R}^n\times(0,\infty).$$

To impose the fixed boundary $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$0, a compact, smooth codimension-2 submanifold, $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$1 is interpreted as the zero-set of a uniformly continuous upper obstacle function $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$2 with

$$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$3

subject to the obstacle constraint

$$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$4

and initial data

$$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$5

The unique viscosity solution to this constrained PDE provides the level-set flow with prescribed boundary (Bian et al., 2023).

2. Viscosity Framework, Existence, and Regularity

The solution is formulated in the viscosity-solution sense following Ishii–Lions–Crandall. For the obstacle problem:

• A viscosity subsolution satisfies, at points where a test function $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$6 touches $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$7 from above and $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$8 (with $$V = H \quad \text{on } \Gamma_t, \qquad \partial \Gamma_t = \Sigma, \qquad \Gamma_0 \text{ given,}$$9),

$V$0

together with $V$1.

• A supersolution is defined dually.
• A viscosity solution is both.

Key properties:

• Existence/Uniqueness: For any bounded, uniformly continuous initial data $V$2 with $V$3, there exists a unique $V$4 solving the level-set PDE with obstacle constraint (Bian et al., 2023).
• Comparison Principle: If $V$5 is a subsolution and $V$6 a supersolution with $V$7, then $V$8 for all $V$9. The principle extends directly to the obstacle setting via doubling-of-variables.
• Regularity: If $H = \operatorname{div}\nu$0 and $H = \operatorname{div}\nu$1 are $H = \operatorname{div}\nu$2-Lipschitz in $H = \operatorname{div}\nu$3, then $H = \operatorname{div}\nu$4 satisfies

$H = \operatorname{div}\nu$5

with exponent $H = \operatorname{div}\nu$6 for mean curvature flow.

3. Consistency with Classical Dirichlet Boundary Mean-Curvature Flow

If the evolving hypersurface family $H = \operatorname{div}\nu$7 is classical and $H = \operatorname{div}\nu$8, solving $H = \operatorname{div}\nu$9 with fixed geometric boundary $\nu$0 for $\nu$1, then the level-set flow with obstacle $\nu$2 yields

$\nu$3

The consistency is established by constructing explicit sub- and supersolutions based on the signed distance function $\nu$4:

• $\nu$5 is a viscosity supersolution with obstacle $\nu$6.
• $\nu$7, for large $\nu$8, is a viscosity subsolution. Via comparison, the zero-levels of the viscosity solution $\nu$9 coincide with those of $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$0, so the interface evolves identically to the smooth case (Bian et al., 2023).

4. Agreement with the Sternberg–Ziemer Dirichlet Flow in Mean-Convex Domains

Let $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$1 be a bounded $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$2 domain with strictly positive inward mean curvature, $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$3 and $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$4. The Sternberg–Ziemer construction produces a unique $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$5 solving

$u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$6

with $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$7 and boundary condition $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$8 on $u(x,t):\mathbb{R}^n\times[0,\infty)\to\mathbb{R}$9. Denote $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$0. Then, the level-set flow with obstacle $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$1 coincides with $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$2 for all $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$3.

The connection is demonstrated by:

• Constructing an upper obstacle $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$4 in $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$5, extended so that $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$6 outside $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$7.
• Comparing the obstacle problem solution $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$8 with known subsolutions shows $\Gamma_t = \{x\in\mathbb{R}^n : u(x,t) = 0\},$9 outside $u < 0$0 and $u < 0$1 inside $u < 0$2.
• By using a renormalization lemma, a reverse comparison $u < 0$3 with an admissible $u < 0$4 provides $u < 0$5. The arguments yield full equivalence of the interface evolution (Bian et al., 2023).

5. Barrier Arguments and Curvature Formulas

Multiple barrier constructions play a central role in ensuring regularity and comparison principles:

• Initial-layer barriers:

$u < 0$6

supporting time-Hölder continuity.

• Boundary-layer subsolutions:

$u < 0$7

for strictly mean-convex $u < 0$8, with corresponding positivity of mean curvature $u < 0$9.

• Push-in domains: Subdomains $u > 0$0 with strictly positive mean curvature on $u > 0$1, constructed by inward perturbations of $u > 0$2.

The evolution of curvature under normal translation is central to these arguments. For $u > 0$3, a $u > 0$4 codimension-$u > 0$5 submanifold, parallel hypersurfaces at distance $u > 0$6 have principal curvatures

$u > 0$7

with an additional direction having curvature $u > 0$8. For small $u > 0$9, the parallel hypersurface $$u_t - |\nabla u|\operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) = 0, \quad \text{in } \mathbb{R}^n\times(0,\infty).$$0 is strictly mean-convex if $$u_t - |\nabla u|\operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) = 0, \quad \text{in } \mathbb{R}^n\times(0,\infty).$$1 is without boundary. This underpins the construction of sub- and supersolutions $$u_t - |\nabla u|\operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) = 0, \quad \text{in } \mathbb{R}^n\times(0,\infty).$$2 and $$u_t - |\nabla u|\operatorname{div}\left(\frac{\nabla u}{|\nabla u|}\right) = 0, \quad \text{in } \mathbb{R}^n\times(0,\infty).$$3 that trap the viscosity solution and enforce coincidence of zero-level sets (Bian et al., 2023).

6. Global Well-Posedness and Relation to Prior Level-Set Frameworks

Integration of the obstacle interpretation, viscosity solution machinery, and barrier techniques allows for global-in-time, well-posed level-set mean curvature flow with prescribed boundaries of arbitrary geometry. The method unifies the level-set approach with both classical boundary-value evolutions and the Sternberg–Ziemer Dirichlet flow, providing a robust theoretical foundation for interface evolution under geometric constraints. This construction demonstrates sharp consistency between the obstacle-based level-set formulation and established mean curvature flow theories (Bian et al., 2023).