Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions
The paper is concerned with the motion of a front \((\Gamma_t)_{t\in [0,T]}\) according to the law \[ v= f(x,t,n,Dn,\Omega_t)\text{ on }\Gamma_t. \] Here, \(v\) is the normal velocity of \(\Gamma_t\), \(n\) the unit outward normal and \(\Omega_t\) the interior of \(\Gamma_t\). The front evolves in a bounded domain \(U\subset \mathbb{R}^N\) and meets \(\partial U\) orthogonally. Using the level set approach the author obtains a unique generalized solution in the form \(\Gamma_t= \{x\in\overline U\mid u(x,t)= 0\}\), where \(u\) is the viscosity solution of a nonlocal parabolic PDE with a Neumann boundary condition. Furthermore, threshold dynamics type approximation schemes for the above evolution law are introduced and their convergence to the level set solution is proved.
scientific article; zbMATH DE number 1880088