Existence of smooth solutions to a one-dimensional nonlinear degenerate variational wave equation

Abstract

This paper is focused on a one-dimensional nonlinear variational wave equation which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. We establish the global existence of smooth solutions to its degenerate initial–boundary value problem under relaxed conditions on the initial–boundary data. Moreover, we show that the solution is uniformly C1,α continuous up to the degenerate boundary and the degenerate curve is C1,α continuous for α∈(0,12).