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We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set Γ o a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

We demonstrate that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure µ + = f + dx onto µ − = f − dy can be constructed by studying the p-Laplacian equationThe idea is to show u p → u, where u satisfiesfor some density a ≥ 0, and then to build a flow by solving an ODE involving a, Du, f + and f − .

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We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a model for phase transition in polycrystalline material. We rigourously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.