A stability property for the generalized mean curvature flow equation
The author studies stability properties for interfaces generated by viscosity solutions of geometric equations. An equation \(u_t-F(x,t,Du,D^2u)=0\) for \((x,t)\in \mathbb{R}^n\times(0,\infty)\) and initial datum \(u(x,0)=u_0(x)\) is called geometric, if the function \(F:\mathbb{R}^n\times \mathbb{R}^+\times \mathbb{R}^n\times S^n\to \mathbb{R}\) satisfies the condition \(F(x,t,\lambda p, \lambda X+\mu(p\otimes p)) =\lambda F(x,t,p,X)\) for any \(\lambda>0\), \(\mu\in \mathbb{R}\), \((x,t,p,X)\in \mathbb{R}^n\times \mathbb{R}^+\times \mathbb{R}^n\times S^n\). A typical example for a geometric equation is the mean curvature flow equation. If the interface is regular, the author proves the stability of the signed distance function in front of geometric perturbations of the equation.
stability of the signed distance function
scientific article; zbMATH DE number 1449034