Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory and a nonlinear du al transformation method for solving the fully nonlinear, non-convex parame tric variational problem inf(W(Lambda u - mu) - F(u)), and associated nonli near boundary value problems, where Lambda is a nonlinear operator, W is ei ther convex or concave functional of p = Lambda u, and mu is a given parame ter. Detailed mathematical proofs are provided for the complementary extrem um principles proposed recently in finite deformation theory. A method for obtaining truly dual variational principles (without a dual gap and involvi ng the dual variable p* of Lambda u only) in n-dimensional problems is prop osed. It is proved that for convex W(p), the critical point of the associat ed Lagrangian L-mu(u, p*) is a saddle point if and only if the so-called co mplementary gap function is positive. In this case, the system has only one dual problem. However, if this gap function is negative, the critical poin t of the Lagrangian is a so-called super-critical point, which is equivalen t to the Auchmuty's anomalous critical point in geometrically linear system s. We discover that, in this case, the system may have more than one primal -dual set of problems. The critical point of the Lagrangian either minimize s or maximizes both primal and dual problems. An interesting triality theor em in non-convex systems is proved, which contains a minimax complementary principle and a pair of minimum and maximum complementary principles. Appli cations in finite deformation theory are illustrated. An open problem left by Hellinger and Reissner is solved completely and a pure complementary ene rgy principle is constructed. It is proved that the dual Euler-Lagrange equ ation is an algebraic equation, and hence, a general analytic solution for non-convex variational-boundary value problems is obtained. The connection between nonlinear differential equations and algebraic geometry is revealed .