Homogenization of a Hamilton–Jacobi equation associated with the geometric motion of an interface

Abstract

This paper studies the overall evolution of fronts propagating with a normal velocity that depends on position, vn = f (x), where f is rapidly oscillating and periodic. A level-set formulation is used to rewrite this problem as the periodic homogenization of a Hamilton{Jacobi equation. The paper presents a series of variational characterization (formulae) of the e® ective Hamiltonian or e® ective normal velocity. It also examines the situation when f changes sign.

Effective motion of a curvature-sensitive interface through a heterogeneous medium

“…This section considers the homogenization of (1.3) assuming that c > 0. The case c = 0 when (1.3) reduces to a Hamilton-Jacobi equation has been treated in [11,15,16,26]. It has been shown that the viscosity solution of the Hamilton-Jacobi initial value problem…”

“…forf determined by the solution of a suitable 'unit cell' problem. Various variational characterizations forf are given in [11]. We will in fact consider the homogenization of a slightly more general problem than (1.3).…”

“…In other words, the curvature term dominates so that the interface becomes flat and we obtain a one-dimensional problem studied in [5]. Similarly, if we assume that the curvature coefficient is smaller, or ε α c for α > 1, then the interface does not feel the contribution of the curvature and one obtains geometric motion studied in [11].…”

“…This paper is the second in a series dealing with the propagation of fronts through heterogeneous media. The first [11] considered only f and ignored the curvature. The third [12] considers the situation where the evolution of the interface is coupled to a partial differential equation in Ω (specifically elasticity).…”

Stochastic homogenization of interfaces moving with changing sign velocity

“…The homogenization of (1.1) in the periodic setting was established by Cardaliaguet, Lions and Souganidis [17]. Prior to their work, Craciun and Bhattacharya [18] discussed formally and gave numerical examples for periodic homogenization of such problems.…”