Pulsating wave for mean curvature flow in inhomogeneous medium

Abstract

We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in an inhomogeneous medium, represented by a periodic forcing. The main difficulty is caused by the degeneracy of the equation and the fact the forcing is allowed to change sign. Under the assumption of weak inhomogeneity, we obtain uniform oscillation and gradient bounds so that the evolving surface can be written as a graph over a reference hyperplane. The existence of an effective speed of propagation is established for any normal direction. We further prove the Lipschitz continuity of the speed with respect to the normal and various stability properties of the pulsating wave. The results are related to the homogenisation of mean curvature flow with forcing.

Bump solutions for the mesoscopic Allen–Cahn equation in periodic media

“…∀x ∈ R n and k ∈ Z n The functional in (1.3) has been considered in [5,11,12] as a mesoscopic model for phase transitions (see also [7,8] for the analysis of the gradient flow of (1.3)).…”

Curve shortening flow in heterogeneous media

“…In [6] existence of traveling wave solutions for (57) has been established. Moreover in [13] (see also [5]) the authors discuss the uniqueness of traveling fronts and characterize the asymptotic speed in some particular case.…”

“…When the forcing term is periodic, equation (1) was recently considered in [6], where the authors prove existence and uniqueness of planar pulsating waves in every direction of propagation. This result leads to the homogenization of (2) for plane-like initial data (see Section 3).…”

“…In particular, in [6,Thm. 4.1] it is shown thatû ε can be represented aŝ u α,ε (x, t) = αx + c(α, ε) 1 + α 2 t + O(ε) ∀(x, t) ∈ R 2 ,…”

“…Notice that, by [6,Prop. 4.4], for all (x, t) ∈ R 2 we have c(α, ε) = 0 ⇒ (û α,ε ) t = 0, c(α, ε) > 0 ⇒ (û α,ε ) t > 0, c(α, ε) < 0 ⇒ (û α,ε ) t < 0.…”