On symmetric div-quasiconvex hulls and divsym-free $L^\infty$-truncations

Behn, Linus; Gmeineder, Franz; Schiffer, Stefan

doi:10.4171/aihpc/66

Cited by 3 publications

(4 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…While this general case is out of reach with the results of this paper, the technique of constructing corrector terms does not seem to be entirely hopeless. Meanwhile, similar to [5,Conjecture 6.4], we conjecture that a truncation result á la 1.1 is possible for some A , whenever A satisfies the complex constant rank condition (i.e. the Fourier symbol has constant rank over all complex Fourier modes).…”

Section: General Differential Constraintssupporting

confidence: 53%

“…Consequently, Lipschitz truncation of a function u under a constraint A u = 0 is closely connected to the L ∞ -truncation of w (= Du) under some constraint Bw = 0. As mentioned before, the latter has been examined in [5,32] in greater detail; in particular (as above) the authors in [5] have conjectured that a low-regularity truncation is possible whenever B obeys the complex constant rank condition.…”

Section: Lower Regularitymentioning

confidence: 95%

“…L ∞ -instead of W 1,∞ ), cf. [5,32] for a discussion of (T2b') and [21,22] for a discussion of (T2c') in that setting.…”

Section: Solenoidal Truncation and The Main Statementmentioning

confidence: 99%

“…First of all, in [9, Section 4] (see also [5,21] for related discussions), the truncation is obtained by writing a divergence-free function u ∈ W 1,p (R 3 ; R 3 ) as…”

Section: Potential Truncationmentioning

confidence: 99%

See 3 more Smart Citations

An alternative approach to solenoidal Lipschitz truncation

Schiffer

2024

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

In this work, we present an alternative approach to obtain a solenoidal Lipschitz truncation result in the spirit of D. Breit, L. Diening and M. Fuchs [Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Equ. 253 (2012), 1910–1942.]. More precisely, the goal of the truncation is to modify a function $u \in W^{1,p}(\mathbb {R}^N;\mathbb {R}^N)$ that satisfies the additional constraint $\operatorname {div} u=0$ , such that its modification $\tilde {u}$ is Lipschitz continuous and divergence-free. This approach is different to the approaches outlined in the aforementioned work and D. Breit, L. Diening and S. Schwarzacher [Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013), 2671–2700, Section 4] and is able to obtain the rather strong bound on the difference between $u$ and $\tilde {u}$ from the former article. Finally, we outline how the approach pursued in this work may be generalized to closed differential forms.

show abstract

Section: General Differential Constraintssupporting

confidence: 53%

Section: Lower Regularitymentioning

confidence: 95%

“…L ∞ -instead of W 1,∞ ), cf. [5,32] for a discussion of (T2b') and [21,22] for a discussion of (T2c') in that setting.…”

Section: Solenoidal Truncation and The Main Statementmentioning

confidence: 99%

“…First of all, in [9, Section 4] (see also [5,21] for related discussions), the truncation is obtained by writing a divergence-free function u ∈ W 1,p (R 3 ; R 3 ) as…”

Section: Potential Truncationmentioning

confidence: 99%

See 2 more Smart Citations

An alternative approach to solenoidal Lipschitz truncation

Schiffer

2024

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

show abstract

Which Measure-Valued Solutions of the Monoatomic Gas Equations are Generated by Weak Solutions?

Gallenmüller

Wiedemann

2023

Arch Rational Mech Anal

View full text Add to dashboard Cite

Contrary to the incompressible case, not every measure-valued solution of the compressible Euler equations can be generated by weak solutions or a vanishing viscosity sequence. In the present paper we give sufficient conditions on an admissible measure-valued solution of the isentropic Euler system to be generated by weak solutions. As one of the crucial steps we prove a characterization result for generating $${\mathcal {A}}$$ A -free Young measures in terms of potential operators including uniform $$L^{\infty }$$ L ∞ -bounds. More concrete versions of our results are presented in the case of a solution consisting of two Dirac measures. We conclude by discussing that are also necessary conditions for generating a measure-valued solution by weak solutions or a vanishing viscosity sequence and will point out that the resulting gap mainly results from obtaining only uniform $$L^p$$ L p -bounds for $$1<p<\infty $$ 1 < p < ∞ instead of $$p=\infty $$ p = ∞ .

show abstract

On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_{3}$ Structure

2023

View full text Add to dashboard Cite

In this article we study quantitative rigidity properties for the compatible and incompatible two-state problems for suitable classes of $\mathcal{A}$ A -free differential inclusions and for a singularly perturbed $T_{3}$ T 3 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 $\epsilon ^{\frac{2}{3}}$ ϵ 2 3 -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); Palombaro and Ponsiglione (Asymptot. Anal. 40(1):37–49, 2004), we discuss the scaling behavior of a $T_{3}$ T 3 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.

show abstract

On symmetric div-quasiconvex hulls and divsym-free $L^\infty$-truncations

Abstract: We establish that for any non-empty, compact set K\subset\mathbb{R}_{\operatorname{sym}}^{3\times 3} the 1 - and \infty -symmetric div-quasiconvex hulls \smash{K^{(1)}} and \smash{K^{(\infty)}} coincide. This settles a conjecture in a recent work … Show more

Cited by 3 publications

References 0 publications

An alternative approach to solenoidal Lipschitz truncation

An alternative approach to solenoidal Lipschitz truncation

Which Measure-Valued Solutions of the Monoatomic Gas Equations are Generated by Weak Solutions?

On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_{3}$ Structure

Contact Info

Product

Resources

About