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Motion of a graph by convexified energy. (English) Zbl 0824.35051
The author studies the evolution of hypersurfaces $$\Gamma_ t$$ in $$\mathbb{R}^ n$$. The equation he considers is a mathematical model for the dynamics of surfaces of a melting solid when the effect outside the surface is negligible. This is a generalization of the famous mean curvature flow. In the case that the so-called interface energy density $$\gamma$$ is not convex it turns out that the equation is not well-posed even locally. Therefore $$\gamma$$ is replaced by its convexification $$\widetilde \gamma$$. Here, the author treats the case of a curve $$(n = 2)$$ given by the graph of a function on $$\mathbb{R}$$. Since $$\widetilde \gamma$$ may not be strictly convex $$\Gamma_ t$$ may develop singularities in finite time. Moreover, $$\widetilde \gamma$$ may no longer be $$C^ 2$$ (outside the origin) even if $$\gamma$$ is smooth. Therefore, the equation has to be understood in a weak sense. Then, the theory of viscosity solutions is applied to show the unique global-in-time existence of solutions $$\Gamma_ t$$ for given initial data $$\Gamma_ 0$$ when $$\Gamma_ 0$$ is represented as the graph of a function growing at most linearly at infinity. An important ingredient in the proof is a comparison principle.

##### MSC:
 35K55 Nonlinear parabolic equations 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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