On scaling properties for two-state problems and for a aingularly perturbed T3 structure

• Verfasst von: Raiƫă, Bogdan [VerfasserIn]
• Verfasst von: Rüland, Angkana [VerfasserIn]
• Verfasst von: Tissot, Camillo [VerfasserIn]
• Titel: On scaling properties for two-state problems and for a aingularly perturbed T3 structure
• Verf.angabe: Bogdan Raiţă, Angkana Rüland, Camillo Tissot
• E-Jahr: 2023
• Jahr: 17 March 2023
• Umfang: 50 S.
• Fussnoten: Im Titel ist die Zahl 3 tiefgestellt ; Gesehen am 24.04.2023
• Titel Quelle: Enthalten in: Acta applicandae mathematicae
• Ort Quelle: [S.l.] : Proquest, 1983
• Jahr Quelle: 2023
• Band/Heft Quelle: 184(2023) vom: März, Artikel-ID 5, Seite 1-50
• ISSN Quelle: 1572-9036
• DOI: doi:10.1007/s10440-023-00557-7
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 35F05
• Sach-SW: 35Q74
• Sach-SW: 74G99
• Sach-SW: 74N05
• Sach-SW: AA-Free inclusions
• Sach-SW: Divergence T3T3
• Sach-SW: Phase transformation
• Sach-SW: Two-well problem
• K10plus-PPN: 1843406799
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of A-free differential inclusions and for a singularly perturbed T3 structure for the divergence operator. In particular, in the compatible setting of the two-state problem we prove that all homogeneous, first order, linear operators with affine boundary data which enforce oscillations yield the typical ϵ23-lower scaling bounds. As observed in Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015) for higher order operators this may no longer be the case. Revisiting the example from Chan and Conti (Math. Models Methods Appl. Sci. 25(06):1091–1124, 2015), we show that this is reflected in the structure of the associated symbols and that this can be exploited for a new Fourier based proof of the lower scaling bound. Moreover, building on Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022); Garroni and Nesi (Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460(2046):1789–1806, 2004, https://doi.org/10.1098/rspa.2003.1249 [Titel anhand dieser DOI in Citavi-Projekt übernehmen] ); Palombaro and Ponsiglione (Asymptot. Anal. 40(1):37–49, 2004), we discuss the scaling behavior of a T3 structure for the divergence operator. We prove that as in Rüland and Tribuzio (Arch. Ration. Mech. Anal. 243(1):401–431, 2022) this yields a non-algebraic scaling law.