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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional
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by Massimo Gobbino and Nicola Picenni
Trans. Amer. Math. Soc. 376 (2023), 5307-5375
DOI: https://doi.org/10.1090/tran/8841
Published electronically: May 9, 2023

Abstract:

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient.

We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale.

Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.

References
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Bibliographic Information
  • Massimo Gobbino
  • Affiliation: Dipartimento di Matematica, Università degli Studi di Pisa, PISA, Italy
  • MR Author ID: 360282
  • ORCID: 0000-0001-9053-3393
  • Email: massimo.gobbino@unipi.it
  • Nicola Picenni
  • Affiliation: Scuola Normale Superiore, PISA, Italy
  • MR Author ID: 1291937
  • Email: nicola.picenni@sns.it
  • Received by editor(s): May 25, 2022
  • Received by editor(s) in revised form: September 16, 2022
  • Published electronically: May 9, 2023
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5307-5375
  • MSC (2020): Primary 49J45, 35B25, 49Q20
  • DOI: https://doi.org/10.1090/tran/8841
  • MathSciNet review: 4630747