  This paper presents a set of complete solutions of a nonconvex variational
problem with a double-well potential. Based on the canonical duality-triality
theory, the associated nonlinear differential equation with either
Dirichlet/Neumann or mixed boundary conditions can be converted into an
algebraic equation, which can be solved analytically to obtain all solutions in
the dual space. Both global and local extremality conditions are identified by
the triality theory. In the application part, typical mechanical models with
specificsources and boundary conditions in $\mathbb{R}^2$ are exhibited.
