Equivalence of weak solution concepts for mean curvature flow

TL;DR

The paper proves a converse to earlier results by showing that if almost every level set of a well-prepared initial datum evolves as both a BV/distributional solution and a Brakke flow, then the evolution is the unique viscosity solution to the level-set mean curvature flow. It constructs a layer-cake function from sublevel evolutions and leverages an L1-continuity plus the Brakke avoidance principle to derive a comparison for variational solutions, leading to equivalence with the viscosity solution. Consequently, almost every level set of a viscosity solution is a variational solution, establishing a robust equivalence between gradient-flow based and comparison-based weak formulations. The work also proves generic uniqueness of variational solutions, showing that non-uniqueness can occur only on nongeneric level sets, and thus provides a unified framework for mean curvature flow across singularities.

Abstract

We provide a connection between weak solution concepts of mean curvature flow. On the one side we have the viscosity solution which is based on the comparison principle. On the other, variational solutions, which are combined Brakke flows and distributional solutions. We prove that if one has a foliation by variational solutions, then the resulting function is the unique viscosity solution. This answers an open question suggested by the work of Evans and Spruck [J. Geom. Anal., 5(1), 1995] and the authors [J. Geom. Anal., 34(12), 2024]. These results show that almost every level set of the viscosity solution is a variational solution. Thus, we establish the equivalence of these solution concepts. Moreover, we show the generic uniqueness of variational solutions.

Figures (3)

• Figure 1: A sliced view of the example $u(x,t)$ in Equation \ref{['Eq:Exu']} for times $t=0,t<1$ and $t\geq 1$.
• Figure 2: The lemniscate, an example of a curve which admits a non-unique variational solution in $\mathbb{R}^2$.
• Figure 3: Visualization of the interpolation construction.

Theorems & Definitions (25)

• Theorem : Reverse direction -- equivalence of solutions
• Definition 2.1: Level set mean curvature flow
• Remark
• Definition 2.2: Viscosity solution
• Definition 2.3: $\mathop{\mathrm{BV}}\limits$ solution, distributional solution, cf. luckhaus1995implicit
• Definition 2.4: Brakke flow
• Definition 2.5: Well-prepared initial datum ESIV
• Remark
• Theorem 2.6: Viscosity solutions are distributional solutions, genericMCF
• Theorem 2.7: Viscosity solutions are Brakke flows ESIV