Long time existence of solutions to an elastic flow of networks

Abstract

TheL(2)-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.