Complementary Duality Research Articles

Duality for Families of Natural Variational Principles in Nonlinear Electrostatics

Standard duality theory embeds a given minimisation problem into a family of such problems and relates the solution of the original problem to a particular member of the dual family of problems. This theory is extended, in the specific context of the nonlinear electrostatic theory of a composite dielectric material, by considering a family of minimisation problems, parametrised by the imposed data, whose solutions define a convex function of this data. Here, the primary variable is the electric field. A corresponding family of dual problems is constructed, in which the primary variable is the electric displacement. The novelty of the formulation is that the duality is interpreted in terms of the space of the parameters that define the original family, so that the dual family generates a convex function of a dual set of parameters. It is demonstrated, under mild hypotheses on the electric properties of the nonlinear dielectric material, that these two convex functions are Legendre transforms of each other. As a by-product, the precise complementary duality principle between individual pairs of variational problems (as opposed to families of variational problems) is elucidated.

A note on complementary modules, duality and reflexiveness

A note on complementary modules, duality and reflexiveness