Periodic homogenization of geometric equations without perturbed correctors

Proving homogenization has been a subtle issue for geometric equations due to the discontinuity when the gradient vanishes. A sufficient condition for periodic homogenization using perturbed correctors is suggested in the literature Caffarelli and Monneau (Arch. Ration. Mech. Anal. 212:503–574, 2014) to overcome this difficulty. However, some noncoercive equations do not satisfy this condition. In this note, we prove homogenization of geometric equations without using perturbed correctors, and therefore we conclude homogenization for the noncoercive equations. Also, we provide a rate of periodic homogenization of coercive geometric equations by utilizing the fact that they remain coercive under perturbation. We also present an example that homogenizes with a rate slower than \(O(\varepsilon )\) $O(\epsilon )$ .