I have the following issue regarding motion by mean curvature, but I suspect it's more an issue of understanding the different notions of the gradient and the divergence of a vectorfield.
We are looking at a hypersurface $\Omega \subset \mathbb{R}^n$ which evolves in time under the mean curvature flow, meaning that its normal velocity is given by the mean curvature of the surface in that point. Now I've found in one source that for motion by mean curvature the velocity can be written as $V=trace(Dn)$ where $n$ denotes the unit normal vector to the surface and $Dn$ its gradient. This was found in the paper of Ishii, Pires and Souganidis: https://projecteuclid.org/euclid.jmsj/1213108018 In another source I've seen that we denote the normal velocity by $V=div(Dn)$. This was found in the paper of Yip and Misiats: https://www.aimsciences.org/article/doi/10.3934/dcds.2016076 The gradient of the vectorfield $n$ is a matrix, so $trace(Dn)$ makes sense. But what is meant by the notation $div(Dn)$? I don't know how to apply divergence to a matrix and I haven't found any reference explaining this.