Two-by-two upper triangular matrices and Morrey’s conjecture

Harris, Terence L. J.; Kirchheim, Bernd; Lin, Chun Chi

doi:10.1007/s00526-018-1360-8

Cited by 9 publications

(7 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We further remark that also in more qualitative arguments related to compensated compactness and Morrey's conjecture, strong, optimal bounds on Riesz transforms have played a major role; see [44] and also [30].…”

Section: Lemma 1 Let Ementioning

confidence: 95%

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

Rüland

Tribuzio

2021

Arch Rational Mech Anal

View full text Add to dashboard Cite

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”.

show abstract

Section: Lemma 1 Let Ementioning

confidence: 95%

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

Rüland

Tribuzio

2021

Arch Rational Mech Anal

View full text Add to dashboard Cite

show abstract

“…However, for n 3, these three notions of convexity are generally not equivalent [20,42]. For n = 2, on the other hand, it is still an open question whether rank-one convexity implies quasiconvexity [6,8,13,18,21,31,[33][34][35]39]. In fact, many classes of functions on R 2×2 have been identified for which rank-one convexity even implies polyconvexity [7,19,25,26,32] and thus quasiconvexity.…”

Section: W ((1 − T)f + T(f + H))mentioning

confidence: 99%

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

Voss¹,

Ghiba

Martín

et al. 2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\textrm {GL}^{\!+}(2)$ of invertible $2\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\textrm {GL}^{\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\colon \textrm {GL}^{\!+}(2)\to \mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

show abstract

“…More recently, the problem of Morrey's conjecture has been approached from the point of view of gradient Young measures and laminates [36,37,38,29,28,32]. In the following, we give a brief overview of this alternative approach and its relation to the optimization methods used above.…”

Section: Gradient Young Measures and Laminatesmentioning

confidence: 99%

Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

Voss,

Martin,

Sander

et al. 2022

Preprint

View full text Add to dashboard Cite

Deciding whether a given function is quasiconvex is generally a difficult task. Here, we discuss a number of numerical approaches that can be used in the search for a counterexample to the quasiconvexity of a given function W . We will demonstrate these methods using the planar isotropic rank-one convex functionwhere λmax ≥ λmin are the singular values of F , as our main example. In a previous contribution, we have shown that quasiconvexity of this function would imply quasiconvexity for all rank-one convex isotropic planar energies W : GL + (2) → R with an additive volumetric-isochoric split of the formwith a concave volumetric part. This example is therefore of particular interest with regard to Morrey's open question whether or not rank-one convexity implies quasiconvexity in the planar case.

show abstract

Two-by-two upper triangular matrices and Morrey’s conjecture

Abstract: It is shown that every homogeneous gradient Young measure supported on matrices of the form a 1,1 · · · a 1,n−1 a 1,n 0 · · · 0 a 2,n is a laminate. This is used to prove the same result on the 3-dimensional nonlinear submanifold of M 2×2 defined by det X = 0 and X 12 > 0.

Cited by 9 publications

References 15 publications

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

On the Energy Scaling Behaviour of a Singularly Perturbed Tartar Square

A rank-one convex, nonpolyconvex isotropic function on with compact connected sublevel sets

Numerical approaches for investigating quasiconvexity in the context of Morrey's conjecture

Contact Info

Product

Resources

About