Radial symmetry and symmetry breaking for some interpolation inequalities

We consider symmetry properties of minimizers in the variational characterization of the best constant in the trace inequality [Formula: see text] in the ball Bρ of radius ρ. When p is fixed, minimizers in this problem can be radial or non-radial depending on the parameters q and ρ. We prove that there is a global radial function u0 > 0, with u0 independent of q, such that any radial minimizer is a multiple of the restriction of u0 to Bρ. Next, we prove that if either q or ρ is sufficiently large, then the minimizers are non-radial. In the case when p = 2, we consider a generalization of the minimization problem and improve some of the above symmetry results. We also present some numerical results describing the exact values of q and ρ for which radial symmetry breaking occurs.

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Improved Interpolation Inequalities and Stability

Advanced Nonlinear Studies ◽

10.1515/ans-2020-2080 ◽

2020 ◽

Vol 20 (2) ◽

pp. 277-291

Author(s):

Jean Dolbeault ◽

Maria J. Esteban

Keyword(s):

Euclidean Space ◽

Symmetry Breaking ◽

Stereographic Projection ◽

Optimal Interpolation ◽

Stability Estimates ◽

Interpolation Inequalities ◽

Carré Du Champ ◽

Optimal Constants ◽

Optimal Functions

AbstractFor exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

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Symmetry breaking in the minimization of the second eigenvalue for composite membranes

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821051300098x ◽

2015 ◽

Vol 145 (1) ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

Claudia Anedda ◽

Fabrizio Cuccu

Keyword(s):

Boundary Condition ◽

Eigenvalue Problem ◽

Symmetry Breaking ◽

Arbitrary Function ◽

Dirichlet Boundary Condition ◽

Dirichlet Boundary ◽

Composite Membranes ◽

Radial Symmetry ◽

Connected Set ◽

Second Eigenvalue

Let Ω ⊂ ℝN be an open bounded connected set. We consider the eigenvalue problem –Δu = λρu in Ω with Dirichlet boundary condition, where ρ is an arbitrary function that assumes only two given values 0 < α < β and is subject to the constraint ∫Ωρ dx = αγ + β(|Ω| – γ) for a fixed 0 < γ < |Ω|. Cox and McLaughlin studied the optimization of the map ρ ⟼ λk(ρ), where λk is the kth eigenvalue. In this paper we focus our attention on the case when N ≥ 2, k = 2 and Ω is a ball. We show that, under suitable conditions on α, β and γ, the minimizers do not inherit radial symmetry.

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A constrained variational model for radial symmetry breaking

Mathematical journal of Ibaraki University ◽

10.5036/mjiu.45.15 ◽

2013 ◽

Vol 45 (0) ◽

pp. 15-31

Author(s):

Shinya Watanabe

Keyword(s):

Symmetry Breaking ◽

Radial Symmetry ◽

Variational Model

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Symmetry breaking in convergent-beam diffraction

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100154378 ◽

1989 ◽

Vol 47 ◽

pp. 480-481

Author(s):

D.J. Eaglesham

Keyword(s):

Symmetry Breaking ◽

Point Group ◽

X Ray Diffraction ◽

Multiple Diffraction ◽

Space Groups ◽

Atomic Displacements ◽

Small Probe ◽

Phase Sensitive ◽

Convergent Beam

Convergent Beam Electron Diffraction is now almost routinely used in the determination of the point- and space-groups of crystalline samples. In addition to its small-probe capability, CBED is also postulated to be more sensitive than X-ray diffraction in determining crystal symmetries. Multiple diffraction is phase-sensitive, so that the distinction between centro- and non-centro-symmetric space groups should be trivial in CBED: in addition, the stronger scattering of electrons may give a general increase in sensitivity to small atomic displacements. However, the sensitivity of CBED symmetry to the crystal point group has rarely been quantified, and CBED is also subject to symmetry-breaking due to local strains and inhomogeneities. The purpose of this paper is to classify the various types of symmetry-breaking, present calculations of the sensitivity, and illustrate symmetry-breaking by surface strains.CBED symmetry determinations usually proceed by determining the diffraction group along various zone axes, and hence finding the point group. The diffraction group can be found using either the intensity distribution in the discs

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Centrosome Aurora A gradient ensures single polarity axis in C. elegans embryos

Biochemical Society Transactions ◽

10.1042/bst20200298 ◽

2020 ◽

Vol 48 (3) ◽

pp. 1243-1253 ◽

Cited By ~ 1

Author(s):

Sukriti Kapoor ◽

Sachin Kotak

Keyword(s):

Symmetry Breaking ◽

Cell Fate ◽

Cell Cortex ◽

Axis Formation ◽

Aurora A Kinase ◽

Aurora A ◽

C Elegans ◽

Cell Embryo ◽

Polarity Establishment

Cellular asymmetries are vital for generating cell fate diversity during development and in stem cells. In the newly fertilized Caenorhabditis elegans embryo, centrosomes are responsible for polarity establishment, i.e. anterior–posterior body axis formation. The signal for polarity originates from the centrosomes and is transmitted to the cell cortex, where it disassembles the actomyosin network. This event leads to symmetry breaking and the establishment of distinct domains of evolutionarily conserved PAR proteins. However, the identity of an essential component that localizes to the centrosomes and promotes symmetry breaking was unknown. Recent work has uncovered that the loss of Aurora A kinase (AIR-1 in C. elegans and hereafter referred to as Aurora A) in the one-cell embryo disrupts stereotypical actomyosin-based cortical flows that occur at the time of polarity establishment. This misregulation of actomyosin flow dynamics results in the occurrence of two polarity axes. Notably, the role of Aurora A in ensuring a single polarity axis is independent of its well-established function in centrosome maturation. The mechanism by which Aurora A directs symmetry breaking is likely through direct regulation of Rho-dependent contractility. In this mini-review, we will discuss the unconventional role of Aurora A kinase in polarity establishment in C. elegans embryos and propose a refined model of centrosome-dependent symmetry breaking.

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