Some remarks on the level set flow by anisotropic curvature (Q1566856)

The author shows that a.e. level sets of the viscosity solution of the anisotropic mean curvature flow satisfy a weak form of the flow equation. More precisely, he considers the evolution of the level sets of the viscosity solution of \(u_{t}+F(t,\nabla u,\nabla^{2}u) = 0,\) where \[ F(t,p,Y) =-|p|M \left({p\over |p|}\right) \left(\sum_{i,j=1} ^{n} {\partial^{2} \gamma\over \partial p_{i}\partial p_{j}} (p) Y_{ij}+\beta(t)\right), \] and \(\gamma\) is the anisotropic surface energy density. He shows that for quite general \(\gamma, M\), \(\beta\), and initial conditions, the generalized anisotropic mean curvature is \(L^{2}\) for a.e. level sets, and that the level sets satisfy a ``Brakke-like'' weak variational inequality. This is a partial generalization of \textit{L. C. Evans} and \textit{J. Spruck} [J. Geom. Anal. 5, 79-116 (1995; Zbl 0829.53040)] in which the related differential inequality is derived in the isotropic case, and is used to prove that almost every level set is a unit-density varifold evolving by mean curvature in the sense of Brakke.