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arXiv · 2112.03832

Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms

Adolfo Arroyo-Rabasa·Paolo Bonicatto·Giacomo Del Nin
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Abstract

We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the $\Gamma$-limit, with respect to the $L^1_{\rm loc}$ topology, of a family of $BMO$-type seminorms is given by $\tfrac14$ times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in $L^1_{\rm loc}$ in terms of the boundedness of the $BMO$-type seminorms.

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Adolfo Arroyo-Rabasa, Paolo Bonicatto, Giacomo Del Nin. 2021-12-07. Representation of the total variation as a $\Gamma$-limit of $BMO$-type seminorms. https://arxiv.org/abs/2112.03832

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