6.
This paper presents a reasonably complete duality theory anda nonlinear dual transformation method for solving the fullynonlinear, non-convex parametric variational problem inf{
W(u- µ) - F(u)}, and associated nonlinear boundary valueproblems, where is a nonlinear operator,
W is either convexor concave functional of
p =
u, and µ is a given parameter.Detailed mathematical proofs are provided for the complementaryextremum principles proposed recently in finite deformationtheory. A method for obtaining truly dual variational principles(without a dual gap and involving the dual variable
p* of
uonly) in
n-dimensional problems is proposed. It is proved thatfor convex
W(p), the critical point of the associated Lagrangian
Lµ(u, p*) is a saddle point if and only if the so-calledcomplementary gap function is positive. In this case, the systemhas only one dual problem. However, if this gap function isnegative, the critical point of the Lagrangian is a so-calledsuper-critical point, which is equivalent to the Auchmuty'sanomalous critical point in geometrically linear systems. Wediscover that, in this case, the system may have more than oneprimal-dual set of problems. The critical point of the Lagrangianeither minimizes or maximizes both primal and dual problems.An interesting triality theorem in non-convex systems is proved,which contains a minimax complementary principle and a pairof minimum and maximum complementary principles. Applicationsin finite deformation theory are illustrated. An open problemleft by Hellinger and Reissner is solved completely and a purecomplementary energy principle is constructed. It is provedthat the dual Euler-Lagrange equation is an algebraic equation,and hence, a general analytic solution for non-convex variational-boundaryvalue problems is obtained. The connection between nonlineardifferential equations and algebraic geometry is revealed.
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