CURVE DIFFUSION AND STRAIGHTENING FLOWS WITH FREE BOUNDARY (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

In this announcement paper, we discuss upcoming results on families of immersed curves $gamma$ : (-1, 1)times[0, T)rightarrow mathbb{R}^{2} with free boundary supported on parallel lines {$eta$_{1}, $eta$_{2}} : mathbb{R}rightarrow mathbb{R}^{2} evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition.