Motion of level sets by mean curvature. IV
[For part III see the authors, ibid. 2, No. 2, 121-150 (1992; Zbl 0768.53003).] For a smooth bounded hypersurface \(\Gamma_0 \subset \mathbb{R}^n\) the authors study the mean curvature evolution PDE \[ u_t = \left(\delta_{ij} - {u_{x_i x_j} \over |Du|^2} \right) u_{x_i x_j} \] in \(\mathbb{R}^n \times (0, \infty)\), \(u = g\) in \(\mathbb{R}^n \times \{t = 0\}\) where \(g : \mathbb{R}^n \to \mathbb{R}\) is a suitably chosen smooth bounded function with \(g^{-1} (0) = \Gamma_0\). This equation has a unique weak solution \(u\), and the authors show that almost every level set of \(u\) is a unit-density varifold evolving by mean curvature in Brakke's sense. Thus even though it is not possible to obtain geometric informations on the evolution of \(\Gamma_0\) under the mean curvature flow past singularities, for a ``generic'' hypersurface one obtains a kind of geometric structure for its mean curvature evolution. The basic idea is to solve for each \(\varepsilon > 0\) the approximate problem \[ u^\varepsilon_t = \left( \delta_{i_j} - {u^\varepsilon_{x_i} u^\varepsilon_{x_j} \over |Du^\varepsilon |^2 + \varepsilon^2} \right) u^\varepsilon_{x_i x_j} \] in \(\mathbb{R}^n \times (0, \infty)\), \(u^\varepsilon = g\) in \(\mathbb{R}^n \times \{t = 0\}\). Using the machinery of compensated compactness it is shown that \(|Du^\varepsilon |\to |Du|\) weakly \(*\) in \(L^\infty (\mathbb{R}^n)\) which together with the coarea formula yields that \[ {Du^\varepsilon \over (|Du^\varepsilon|^2 + \varepsilon^2)^{1/2}} \to {Du \over |Du|} \] strongly in \(L^2_{\text{loc}} (\{|Du|> 0\}, \mathbb{R}^n)\). With further careful and technically involved analysis the function \[ H = \begin{cases} u_t/ |Du|& \text{if } |Du |> 0,\\ 0 & \text{if }|Du |= 0\end{cases} \] can be interpreted as mean curvature and \(-H(Du/ |Du|)\) as the mean curvature vector on \(\{|Du|> 0\}\).
scientific article; zbMATH DE number 734767
scientific article