Studies on approximation, regularity and singularity for mean curvature flow

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We consider an approximation scheme for anisotropic curvature flow, arising from the image processing., and prove the convergence of this scheme. We also study the parabolic free boundary problems arising from mathematical finance. We introduce a time-discrete scheme for a parabolic variational inequality and obtain the convergences of approximate solutions and approximate free boundaries. We also prove the existence and uniqueness of solutions and of free boundary for an investment problem.