Instability of stationary solutions for equations of curvature-driven motion of curves (Q1898801)

The authors study the equations of evolving curves on a two-dimensional square domain \(\Omega\). Here, the curve is assumed to move depending on its curvature, normal vector, and position, and to intersect \(\partial \Omega\) orthogonally at its endpoints. The solutions are considered in the viscosity sense. In a special case the curvature equation together with the boundary condition strongly resembles an ODE with homogeneous Neumann boundary conditions. Using this connection, the authors prove the instability of stationary solutions through an eigenvalue analysis, provided certain conditions are satisfied.