On the energy scaling behaviour of a singularly perturbed Tartar square

• Verfasst von: Rüland, Angkana [VerfasserIn]
• Verfasst von: Tribuzio, Antonio [VerfasserIn]
• Titel: On the energy scaling behaviour of a singularly perturbed Tartar square
• Verf.angabe: Angkana Rüland & Antonio Tribuzio
• Jahr: 2022
• Umfang: 31 S.
• Fussnoten: Online veröffentlicht:10. Dezember 2021 ; Gesehen am 24.04.2023
• Titel Quelle: Enthalten in: Archive for rational mechanics and analysis
• Ort Quelle: Berlin : Springer, 1957
• Jahr Quelle: 2022
• Band/Heft Quelle: 243(2022), 1, Seite 401-431
• ISSN Quelle: 1432-0673
• DOI: doi:10.1007/s00205-021-01729-1
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1843460602
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

In this article we derive an (almost) optimal scaling law for a singular perturbation problem associated with the Tartar square. As in Winter (Eur J Appl Math 8(2):185-207, 1997), Chipot (Numer Math 83(3):325-352, 1999), our upper bound quantifies the well-known construction which is used in the literature to prove the flexibility of the Tartar square in the sense of the flexibility of approximate solutions to the differential inclusion. The main novelty of our article is the derivation of an (up to logarithmic powers matching) ansatz free lower bound which relies on a bootstrap argument in Fourier space and is related to a quantification of the interaction of a nonlinearity and a negative Sobolev space in the form of “a chain rule in a negative Sobolev space”. Both the lower and the upper bound arguments give evidence of the involved “infinite order of lamination”.