Tri-Duality in Nonconvex Systems

Abstract

From traditional convex systems to general nonlinear systems, we will study, in this chapter, the generalized duality theory and analytic solutions for one-dimensional nonconvex variational problems with applications to phase transitions, post-bifurcation, nonsmooth elastoplasticity and nonconvex dynamical systems. Because of the nonconvexity, the nice simple symmetry in the governing equations is broken and the beautiful one-to-one global duality relation no longer exists. The solutions in these systems are usually not unique. But, by introducing a suitable nonlinear operator A, a generalized framework may still be established for many systems. The resulting triality theory reveals interesting local dualities (see Fig. 3.1, the associated problem is given in Example 3.3).

Hidden harmony is stronger than the explicit one.

—Heraclitus

Symmetry breaking leads to phenomena.

—Pierre Curie

The shortest path between two truths in the real domain passes through the complex domain.

—Jacques Hadamard