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We analyze two recently proposed methods to establish a priori lower boundson the minimum of general integral variational problems. The methods, whichinvolve either 'occupation measures' [Korda et al., ch. 10 of NumericalControl: Part A, 10.1016/bs.hna.2021.12.010] or a 'pointwise dual relaxation'procedure [Chernyavsky et al., arXiv:2110.03079], are shown to produce the samelower bound under a coercivity hypothesis ensuring their strong duality. Wethen show by a minimax argument that the methods actually evaluate the minimumfor classes of one-dimensional, scalar-valued, or convex multidimensionalproblems. For generic problems, however, we conjecture that these methodsshould fail to capture the minimum and produce non-sharp lower bounds. Weexplain why using two examples, the first of which is one-dimensional andscalar-valued with a non-convex constraint, and the second of which ismultidimensional and non-convex in a different way. The latter exampleemphasizes the existence in multiple dimensions of nonlinear constraints ongradient fields that are ignored by occupation measures, but are built into thefiner theory of gradient Young measures, which we review.

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