On the Optimal Value of a Variational Form

Abstract

A system S, such as a structure, with physical characteristics defined by a measure $\mu $ on S and a performance which is the minimum value, taken over all permissible functions, of an energy expression in the form of an integral with respect to measure $\mu $ is considered. It is proved that the measure $\mu $ that yields the optimal performance and satisfies $\mu ( S ) = c$ has the property that it renders the energy density constant over the system. For a performance expressed by a more complex variational form, the differential with respect to measure $\mu $ of the variational form must be constant over the entire S. These theorems can be used to generalize some of the available results on the optimal design of structures to more than one dimension.