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Pulsating Wave for Mean Curvature Flow in Inhomogeneous Medium
Nicolas Dirr, Georgia Karali and Aaron Nung Kwan Yip
We prove the existence and uniqueness of pulsating waves for the motion by mean curvature of an n-dimensional hypersurface in a periodic 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 its 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. The existence of an effective speed of propagation is established for any normal direction. We further prove the continuity of the speed with respect to the normal and various stability properties of the pulsating wave. This result is related to the homogenization of forced mean curvature flow.