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This paper mainly addresses the extrema of a nonconvex functional withdouble-well potential in higher dimensions through the approach of nonlinearpartial differential equations. Based on the canonical duality method, thecorresponding Euler--Lagrange equation with Neumann boundary condition can beconverted into a cubic dual algebraic equation, which will help find the localextrema for the primal problem. In comparison with the 1D case discussed by D.Gao and R. Ogden, there exists huge difference in higher dimensions, which willbe explained in the theorem.

On the extrema of a nonconvex functional with double-well potential in higher dimensions