Conservative solutions to the first initial-boundary value problem for a nonlinear variational wave equation

This paper concerns with the first initial-boundary value problem for a one-dimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle.We introduce the energy-dependent coordinates into the finite interval to transform the original problem into a boundary value problem of a semilinear system.The global existence of weak solutions for the new boundary value problem is first established by deriving the a priori estimates and then expressed in terms of the original variables.