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**Motion of a set by the curvature of its boundary.**
*(English)*
Zbl 0769.35070

This important paper is devoted to the study of the weak theory for the generalized mean curvature equation. The newly developed theory of viscosity solution is used.

The approach is closely related to that of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck, who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation. The work brings an intrinsic definition, although their results are used in an essential way.

Main results: The existence of a solution, large time asymptotics of this solution, and its connection to the level set solution of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck. Under restrictive assumptions a uniqueness result is established. A class of explicit solutions which are dilations of Wulff crystals are constructed.

The approach is closely related to that of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck, who view the boundary of the crystal as the level set of a solution to a nonlinear parabolic equation. The work brings an intrinsic definition, although their results are used in an essential way.

Main results: The existence of a solution, large time asymptotics of this solution, and its connection to the level set solution of Osher and Sethian, Chen, Giga and Goto, and Evans and Spruck. Under restrictive assumptions a uniqueness result is established. A class of explicit solutions which are dilations of Wulff crystals are constructed.

Reviewer: J.Lovíšek (Bratislava)

### MSC:

35R35 | Free boundary problems for PDEs |

82D25 | Statistical mechanics of crystals |

35K55 | Nonlinear parabolic equations |