and the stored energy W W ( 1 2 e 2 -) = 1 2 E( 1 2 e 2 -) 2 ; E ยฟ 0 (2) is a double-well function of e = u ; x for any given parameter ยฟ 0. The corresponding Euler-Lagrange equation of the problem is then a nonlinear di erential equation
Even for this very simple problem, the analytic solutions for u(x) cannot be well determined by traditional analytic methods. If f = 0; = 1=2, we should have u ; x = {0; ยฑ1} โx โ I: Hence, any zigzag function with slopes 0 and ยฑ1 satisรฟes this equation, but may not be a minimizer of the total potential energy P(u). Therefore, some numerical discretization approaches for this problem have been examined in [50].
The nonconvex variational problem with double-well structure was รฟrst studied by Van der Waals in 1893 for a compressible uid whose free energy at constant temperature depends not only on the density, but also on the density gradient (see [51]). This problem also appears in hysteresis and phase transitions, super-conductivity, cosmology, mathematical economics, nonconvex dynamical systems, nonlinear bifurcation and post-buckling problems of large deformed structures (cf. e.g., [7,8,14,18,26,34,39]).
The direct approaches and relaxation methods for solving nonlinear equilibrium equations have been discussed extensively for more than 30 years (see, for example, [5, 9 -11,13,15,19,21,38,40,41,43,46,47,55]). It is known that in nonlinear variational problems, traditional direct methods can provide only upper bound approaches to the solution. The so-called relaxation method can be used mainly for รฟnding global minimizer of the nonconvex energy. However, in post-bifurcation analysis and phase transitions, local maximizers usually play more important roles. As was indicated in [40], the relaxation method for solving nonconvex variational problems with three or more phases (potential wells) is fundamentally more di cult.
Duality theory for geometrically linear variational problem inf J (u; u), where is a linear di erential operator (say, u = u ; x ), has been well studied for both convex and nonconvex systems (see, for example, [1,3,4,12,16,27,31,44, 46 -49,52,56-61, 63-65]). The dual functional J * ( * e * ; e * ) in these classical principles is usually obtained by Fenchel-Legendre transformation, where e * is the dual variable of e = u. For nonconvex stored energy, to รฟnd the Legendre dual function is very di cult, or even impossible. For example, if W e =W ( (e)) deรฟned by ( 2) is a double-well function of e, the dual variable e * of e deรฟned by e * = @W e (e) @e = e 1 2 e 2is nonlinearly dependent on the deformation rate e = u ; x . The inverse form e(e * ) is very complicated, and therefore, the Legendre-conjugate function W c (e * ) deรฟned by W c e (e * ) = e * e(e * ) -W e (e(e * )) does not have a simple algebraic expression (see [52]). The Fenchel sup-conjugate function, deรฟned by W * e (e * ) = sup e {e * e -W e (e)}