Construction of Lipschitz solutions to an Eikonal type equation on annulus

Motivated by a minimization problem in micromagnetics, we study the Lipschitz solutions to a boundary value problem of an Eikonal type equation on an annulus with a parameter. Exact condition on the parameter is given for the existence of solutions. Under this condition the solutions are constructed by gluing the local characteristic solutions near the boundary with certain special solutions of the Eikonal equation away from the boundary.