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 $\Sigma$, a compact, smooth codimension-2 submanifold, $\Sigma$ is interpreted as the zero-set of a uniformly continuous upper obstacle function $\psi^+(x)\geq0$ with

$\{x: \psi^+(x)=0\} = \Sigma,$

subject to the obstacle constraint

$$0 \leq u(x,t) \leq \psi^+(x), \quad \forall x \in \mathbb{R}^n,\; t>0,$$

and initial data

$$u(\cdot,0)=u_0\in BUC(\mathbb{R}^n), \qquad \{x:u_0(x)=0\} = \Gamma_0,\quad 0 \leq u_0 \leq \psi^+.$$

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 $\varphi\in C^2$ touches $u^*$ from above and $u^*(\hat{x},\hat{t}) > \psi^-(\hat{x},\hat{t})$ (with $\psi^-\equiv 0$),

$$\varphi_t(\hat{x},\hat{t}) - |\nabla \varphi|\,\operatorname{div}\left(\frac{\nabla \varphi}{|\nabla \varphi|}\right)(\hat{x},\hat{t}) \leq 0,$$

together with $0\leq u^*\leq \psi^+$.

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

Key properties:

• Existence/Uniqueness: For any bounded, uniformly continuous initial data $u_0$ with $0\leq u_0\leq \psi^+$, there exists a unique $u\in BUC(\mathbb{R}^n\times[0,\infty))$ solving the level-set PDE with obstacle constraint (Bian et al., 2023).
• Comparison Principle: If $u$ is a subsolution and $v$ a supersolution with $u(\cdot,0)\leq v(\cdot,0)$, then $u\leq v$ for all $t>0$. The principle extends directly to the obstacle setting via doubling-of-variables.
• Regularity: If $\psi^+$ and $u_0$ are $L$-Lipschitz in $x$, then $u$ satisfies

$$|u(x,t) - u(y,t)|\leq L|x-y|,\quad |u(x,t)-u(x,s)|\leq C|t-s|^{1/2},$$

with exponent $1/2$ for mean curvature flow.

3. Consistency with Classical Dirichlet Boundary Mean-Curvature Flow

If the evolving hypersurface family $\Gamma^s_t$ is classical and $C^{2,1}$, solving $V = H$ with fixed geometric boundary $\Sigma$ for $0\leq t \leq T$, then the level-set flow with obstacle $\Sigma$ yields

$$\Gamma_t = \Gamma^s_t,\quad \forall\, 0 \leq t \leq T.$$

The consistency is established by constructing explicit sub- and supersolutions based on the signed distance function $d^s(x,t)=\pm\operatorname{dist}(x,\Gamma_t^s)$:

• $w(x,t) = \min\{|d^s(x,t)|,\,\delta\}$ is a viscosity supersolution with obstacle $\psi^+=\operatorname{dist}(x,\Sigma)$.
• $z(x,t) = \exp(-\lambda t)w(x,t)$, for large $\lambda$, is a viscosity subsolution. Via comparison, the zero-levels of the viscosity solution $u$ coincide with those of $d^s$, 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 $U\Subset \mathbb{R}^n$ be a bounded $C^2$ domain with strictly positive inward mean curvature, $\Gamma_0\setminus \Sigma \subset U$ and $\Sigma\subset\partial U$. The Sternberg–Ziemer construction produces a unique $v\in C(\bar U\times[0,\infty))$ solving

$$v_t - |\nabla v|\,\operatorname{div}\Bigg(\frac{\nabla v}{|\nabla v|}\Bigg) = 0\quad \text{in } U,$$

with $v(\cdot,0)=\operatorname{dist}(\cdot, \Gamma_0)$ and boundary condition $v = \operatorname{dist}(\cdot, \Gamma_0)$ on $\partial U$. Denote $\Gamma_t^U = \{x:v(x,t)=0\}$. Then, the level-set flow with obstacle $\Sigma$ coincides with $\Gamma_t^U$ for all $t$.

The connection is demonstrated by:

• Constructing an upper obstacle $\psi^+(x,t) = v(x,t) + \sigma(x)$ in $\bar U$, extended so that $\psi^+ > 0$ outside $\bar U$.
• Comparing the obstacle problem solution $u$ with known subsolutions shows $u>0$ outside $\bar U$ and $u\leq v$ inside $U$.
• By using a renormalization lemma, a reverse comparison $v\leq \theta(u)$ with an admissible $\theta$ provides $\Gamma_t\subset\Gamma_t^U$. 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:

$$h(x,t)=L\left(\frac{ct}{\delta}+\delta+\frac{|x-x_0|^2}{4\delta}\right)+u_0(x_0),$$

supporting time-Hölder continuity.

• Boundary-layer subsolutions:

$w(x)=\operatorname{dist}(x,U),$

for strictly mean-convex $U$, with corresponding positivity of mean curvature $$\operatorname{div}\big(\frac{\nabla w}{|\nabla w|}\big)>0$$.

• Push-in domains: Subdomains $V\subset U$ with strictly positive mean curvature on $\partial V$, constructed by inward perturbations of $\partial U$.

The evolution of curvature under normal translation is central to these arguments. For $\Sigma\subset \mathbb{R}^n$, a $C^2$ codimension-$k$ submanifold, parallel hypersurfaces at distance $\delta$ have principal curvatures

$$\kappa_i^\delta = \frac{\kappa_i}{1\pm \delta \kappa_i},$$

with an additional direction having curvature $\pm 1/\delta$. For small $\delta$, the parallel hypersurface $\partial U_\delta(\Sigma)$ is strictly mean-convex if $\Sigma$ is without boundary. This underpins the construction of sub- and supersolutions $z(x,t)$ and $w(x,t)$ 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).