Abstract
This paper treats degenerate parabolic equations of second order
related to differential geometry, where ∇ stands for spatial derivatives of u = u{t,x) in x ∈ R n , and u t represents the partial derivative of u in time t. We are especially interested in the case when (1.1) is regarded as an evolution equation for level surfaces of u. It turns out that (1.1) has such a property if F has a scaling invariance
for a nonzero p ∈ R n and a real symmetric matrix X, where ⊗ denotes a tensor product of vectors in R n. We say (1.1) is geometric if F satisfies (1.2). A typical example is
where ∇u is the (spatial) gradiant of u. Here ∇u/|∇u| is a unit normal to a level surface of u, so div (∇u/|∇u|) is its mean curvature unless ∇u vanishes on the surface. Since u t /\∇u\ is a normal velocity of the level surface, (1.3) implies that a level surface of solution u of (1.3) moves by its mean curvature unless ∇u vanishes on the surface. We thus call (1.3) the mean curvature flow equation in this paper.
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Chen, YG., Giga, Y., Goto, S. (1999). Uniqueness and Existence of Viscosity Solutions of Generalized mean Curvature Flow Equations. In: Ball, J.M., Kinderlehrer, D., Podio-Guidugli, P., Slemrod, M. (eds) Fundamental Contributions to the Continuum Theory of Evolving Phase Interfaces in Solids. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59938-5_14
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DOI: https://doi.org/10.1007/978-3-642-59938-5_14
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