Sharpness and non-sharpness of occupation measure bounds for integral variational problems

TL;DR

This paper compares two methods for establishing lower bounds on integral variational problems, showing their equivalence under certain conditions and highlighting their limitations in capturing the true minimum for complex, non-convex problems.

Contribution

It demonstrates the equivalence of occupation measure and dual relaxation methods under strong duality and analyzes their limitations in non-convex, multidimensional problems.

Findings

Methods produce the same bounds under coercivity and duality.

They evaluate the minimum for certain classes of problems.

They fail to capture the minimum in generic non-convex cases.

Abstract

We analyze two recently proposed methods to establish a priori lower bounds on the minimum of general integral variational problems. The methods, which involve either `occupation measures' or a `pointwise dual relaxation' procedure, are shown to produce the same lower bound under a coercivity hypothesis ensuring their strong duality. We then show by a minimax argument that the methods actually evaluate the minimum for classes of one-dimensional, scalar-valued, or convex multidimensional problems. For generic problems, however, these methods should fail to capture the minimum and produce non-sharp lower bounds. We demonstrate this using two examples, the first of which is one-dimensional and scalar-valued with a non-convex constraint, and the second of which is multidimensional and non-convex in a different way. The latter example emphasizes the existence in multiple dimensions of nonlinear constraints on gradient fields that are ignored by occupation measures, but are built into the finer theory of gradient Young measures.