On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

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”.

Optimal fine-scale structures in compliance minimization for a uniaxial load in three space dimensions

“…In the former, there exist regimes of simple hanging drapes with dyadic coarsening structures along essentially just one dimension [19], but also regimes with complicated multidirectional patterns as in crumpling paper [8]. Likewise, in martensites plain dyadic branching patterns occur in the simplest situation [12], while in some shape memory alloys there seems to be enormous flexibility in the possible patterns [16].…”

“…Again by checking that the normal stresses add up to zero at all domain interfaces (except the bottom face of O 12 and the top face of O 16 ) we obtain that σ is divergence-free with normal tensile stress of magnitude F at the top boundary of the boundary cell and normal tensile stress of magnitude 1 on (−s, s) 2 × {0}, where it is attached to the elementary cell underneath, whose stress it exactly balances.…”

“…. , ∂C 4 , while it is given by −2σ rz ( w 16 , ϕ, l − z) = −2 16 w R(l − z)R (l − z) = − 32 w R(z)R (z) on ∂C 0 . Thus, for g 2 (z) = R(z)R (z)…”

Energy Scaling Law for a Singularly Perturbed Four-Gradient Problem in Helimagnetism

“…Pattern formation is then often related to competing terms in the energy functional, favoring rather uniform or highly oscillatory structures, respectively. The proofs of the scaling laws often involve branching-type constructions where structures oscillate on refining scales near the boundary, among many others see for instance the scaling laws in , Conti (2000Conti ( , 2006, Otto (2009, 2012), Chan and Conti (2015), Knüpfer et al (2013), Bella and Goldman (2015), Conti and Zwicknagl (2016), Conti et al (2020), Rüland and Tribuzio (2022)) for martensitic microstructure, (Kohn andWirth 2014, 2016) for compliance minimization, (Choksi et al 2004(Choksi et al , 2008) for type-I-superconductors, (Ben Belgacem et al 2002;Bella and Kohn 2014;Bourne et al 2017;Conti et al 2005) for compressed thin elastic films, (Conti and Ortiz 2016;Conti and Zwicknagl 2016) for dislocation patterns, and (Brancolini and Wirth 2017;Brancolini et al 2018) for transport networks.…”

Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions

“…In this context, the incompatible two-well problem was studied in [17] in which an incompatible two-well analogue of the Friesecke-James-Müller rigidity result [33] was used. Moreover, the study of model singular perturbation problems for the analysis of austenite-martensite interfaces in terms of a surface energy parameter [47,46] laid the basis for an intensive, closely related research on singular perturbation problems for shape-memory alloys [16,65,26,53,27,22,29,19,20,60,63,61]. Contrary to the full nucleation problems, in these settings the phenomenon of compatibility plays the main role, while nucleation phenomena in addition require the analysis of the phenomenon of selfaccommodation.…”