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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.

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