Dynamical stability of various convex graphical translators

In the first part of the paper, we prove the existence of longtime solution to mean curvature flow starting from a graph of a continuous function defined over a slab. Then, we establish dynamical stability results for various types of graphical translators to mean curvature flow, namely the grim reaper, two dimensional graphical translators, and asymptotically cylindrical translators.

💡 Research Summary

The paper addresses two fundamental problems in the theory of mean curvature flow (MCF) for graphical hypersurfaces: (1) the long‑time existence of solutions when the initial data is a continuous function defined on a slab, and (2) the dynamical stability of several important families of graphical translators.

Long‑time existence on a slab.
Let Ωⁿᵇ = ℝⁿ⁻¹ × (−b,b) be a slab in ℝⁿ. For any continuous function u₀ ∈ C⁰(Ωⁿᵇ) the authors prove that there exists a smooth solution u(x,t) of the graphical MCF equation
∂ₜu = √{1+|∇u|²} div(∇u/√{1+|∇u|²})
on Ωⁿᵇ × (0,∞) which attains u₀ as t → 0⁺ and remains continuous up to t = 0. The proof relies on a maximum‑principle argument combined with barrier functions that control the gradient and second derivatives uniformly in time. Moreover, if u₀ grows sufficiently large near the two boundary hyperplanes of the slab (a condition expressed by a uniform lower bound on |u₀| in a thin strip near |xₙ| = b), then each time‑slice inherits the same growth property. This result extends earlier work of Ecker–Huiskens and Clutterbuck, which required locally Lipschitz or C^{2+α} regularity, by showing that mere continuity suffices.

Dynamical stability framework.
A translator is a hypersurface M ⊂ ℝⁿ⁺¹ satisfying ⟨H,ν⟩ = ⟨ν, e_{n+1}⟩; under MCF it moves by translation M_t = M + t e_{n+1}. Dynamical stability asks whether a small perturbation of a translator, taken as a graphical hypersurface, converges back (up to rigid motions) to a translate of the original under the flow. The authors adopt a barrier‑method: two copies of the given translator (shifted up and down) serve as supersolution and subsolution, yielding uniform C⁰ bounds for the perturbed flow. These bounds are upgraded to higher‑order estimates via standard Schauder theory. The key analytical tool for identifying the limiting flow is Hamilton’s Harnack inequality, which forces any eternal limit sandwiched between two translators to be a translator with the same velocity.

Stability of the Grim Reaper (1‑dimensional translator).
The Grim Reaper curve u(x)=−log cos x on (−π/2,π/2) is the unique 1‑dimensional translator. Theorem 1.3 shows that if the initial graph u₀ satisfies a finite C⁰ distance from u, has bounded curvature, and finite total curvature, then the graphical curve‑shortening flow exists for all time and satisfies
lim_{t→∞}