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Multiple solutions for a nonconvex variational problem with double‐well potentials

In this paper, we present a set of complete solutions for a nonconvex variational problem with double‐well potentials in higher dimensions. Based on the canonical duality theory, the corresponding nonlinear Euler‐Lagrange equation with Neumann boundary condition can be converted into an algebraic equation, which can be solved analytically to obtain the solutions of the dual problem. Correspondingly, local extrema of the primal problem can be identified by the pure complementary energy principle.