Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

2011/07/05 by Régis Monneau, Jean-Michel Roquejoffre, Monneau, Régis +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1107.0896

36 pages, 6 figures

arxiv created 2011/07/05 · arxiv updated 2011/07/06

Abstract

We construct travelling wave graphs of the form z=-ct+ϕ(x), ϕ: x ∈ ℝ^N-1 ↦ ϕ(x)∈ ℝ, N ≥ 2, solutions to the N-dimensional forced mean curvature motion V_n=-c₀+κ (c≥ c₀) with prescribed asymptotics. For any 1-homogeneous function ϕ_∞, viscosity solution to the eikonal equation |Dϕ_∞|=√((c/c₀)²-1), we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by ϕ_∞. We also describe ϕ_∞ in terms of a probability measure on \mathbbS^N-2.

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