Fantuzzi G, Tobasco I (2024)
Publication Language: English
Publication Status: In review
Publication Type: Unpublished / Preprint
Future Publication Type: Journal article
Publication year: 2024
URI: https://arxiv.org/abs/2207.13570
Open Access Link: https://arxiv.org/abs/2207.13570
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' [Korda et al., ch. 10 of Numerical Control: Part A, https://doi.org/10.1016/bs.hna.2021.12.010] or a 'pointwise dual relaxation' procedure [Chernyavsky et al., arXiv:2110.03079], 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, we conjecture that these methods should fail to capture the minimum and produce non-sharp lower bounds. We explain why 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, which we review.
APA:
Fantuzzi, G., & Tobasco, I. (2024). Sharpness and non-sharpness of occupation measure bounds for integral variational problems. (Unpublished, In review).
MLA:
Fantuzzi, Giovanni, and Ian Tobasco. Sharpness and non-sharpness of occupation measure bounds for integral variational problems. Unpublished, In review. 2024.
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