Non-parametric mean curvature evolution with boundary conditions
An evolution equation, \(\dot u+Au=0\) in \(\Omega\) \(\times [0,t]\) has previously been investigated where the surfaces, \(m_ t=graph\) u(\(\cdot,t)\) moves in the \(x^{n+1}\)-direction with speed dictated by their mean curvature, \(H=-Au\). This paper replaces this speed to be in the direction of their unit normal equal to the mean curvature. With this focus in mind the paper begins with the boundary value problem, (1) \(\dot u+(1+| Du|^ 2)^{1/2}Au=0\) in \(\Omega\) \(\times [0,\infty]\), (2) \(a^ i(Du)\cdot \gamma_ i=0\) on \(\partial \Omega \times [0,\infty]\), (3), \(u(\cdot,0)=u_ 0\). The operator A in equation (1) is the minimal surface operator given by \[ A=-D_ i(a^ i(p)),\quad a^ i=p^ i(1+| p|^ 2)^{-1\setminus 2} \] and \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with \(\partial \Omega\) being of class, \(C^{2,\alpha}\). The outer unit normal to \(\partial \Omega\) is denoted by \(\gamma =(\gamma_ 1,\gamma_ 2,...,\gamma_ n)\). The paper then continues to prove in a well written organized style two principal theorems. The first one requires that the boundary condition given in equation (3) satisfies \(u_ 0\in C^{2,\alpha}({\bar \Omega})\) and \(a^ i(Du_ 0)\cdot \gamma_ i=0\) on \(\partial \Omega\). Under this hypothesis it is then shown that the problem (1), (2), (3) has a smooth solution u and \(u_ t=u(\cdot,t)\) converges to a constant function as \(t\to \infty\). The proof is elegant and requires several technical details. The other theorem replaces the boundary condition given in equation (2) with \(u_ 0=\phi\) on \(\partial \Omega\) where \(\phi \in C^{2,\alpha}({\bar \Omega})\). In this situation the problem then has again a smooth solution u and \(u_ t=u(\cdot,t)\) converges to the solution of the minimal surface equation with boundary data \(\phi\) as \(t\to \infty\).
scientific article; zbMATH DE number 4124116