scholarly journals Improved Interpolation Inequalities and Stability

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.

scholarly journals On superintegrable symmetry-breaking potentials in N-dimensional Euclidean space

2002 ◽  
Vol 35 (22) ◽  
pp. 4755-4773 ◽  
Author(s):  
E G Kalnins ◽  
G C Williams ◽  
W Miller Jr ◽  
G S Pogosyan
Keyword(s):  
Euclidean Space ◽  
Symmetry Breaking ◽  
Dimensional Euclidean Space

scholarly journals On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

scholarly journals Radial symmetry and symmetry breaking for some interpolation inequalities

2011 ◽  
Vol 42 (3-4) ◽  
pp. 461-485 ◽  
Author(s):  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Gabriella Tarantello ◽  
Achilles Tertikas
Keyword(s):  
Symmetry Breaking ◽  
Radial Symmetry ◽  
Interpolation Inequalities

scholarly journals A relation between the right triangle and circular tori with constant mean curvature in the unit 3-sphere

2004 ◽  
Vol 76 (4) ◽  
pp. 639-643 ◽  
Author(s):  
Abdênago Barros
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Stereographic Projection ◽  
Inverse Image ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
The Right

In this note we will show that the inverse image under the stereographic projection of a circular torus of revolution in the 3-dimensional euclidean space has constant mean curvature in the unit 3-sphere if and only if their radii are the catet and the hypotenuse of an appropriate right triangle.

Operator regularization on the hypersphere

1989 ◽  
Vol 67 (7) ◽  
pp. 669-677 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Euclidean Space ◽  
Sigma Model ◽  
Gauge Theories ◽  
Stereographic Projection ◽  
Mass Parameter ◽  
Field Theories ◽  
Dimensional Euclidean Space ◽  
Two Dimensional ◽  
Yang Mills ◽  
Infrared Cutoff

Operator regularization has proved to be a viable way of computing radiative corrections that avoids both the insertion of a regulating parameter into the initial Lagrangian and the occurrence of explicit infinities at any stage of the calculation. We show how this regulating technique can be used in conjunction with field theories defined on an n + 1-dimensional hypersphere, which is the stereographic projection of n-dimensional Euclidean space. The radius of the hypersphere acts as an infrared cutoff, thus eliminating the need to insert a mass parameter to serve as an infrared regulator. This has the advantage of leaving conformai symmetry present in massless theories, intact. We illustrate our approach by considering [Formula: see text], massless Yang–Mills gauge theories and the two-dimensional nonlinear bosonic sigma model with torsion. In the last model, the lowest mode is used as an infrared cutoff.

Mapping the axial current onto the hypersphere

1990 ◽  
Vol 68 (1) ◽  
pp. 54-57 ◽  
Author(s):  
D. G. C. McKeon
Keyword(s):  
Euclidean Space ◽  
Stereographic Projection ◽  
Axial Current ◽  
Dimensional Euclidean Space ◽  
Gauge Invariant ◽  
Five Dimensions ◽  
Conformally Invariant

A stereographic projection has been used to map conformally invariant spinor electrodynamics from four-dimensional Euclidean space onto the surface of a sphere in five dimensions. We extend electrodynamics so that the spinors also couple in a gauge invariant way to the current on the surface of the hypersphere that is the image of the axial current [Formula: see text] in Euclidean space under this mapping.

scholarly journals Interpolation inequalities in \begin{document}$ \mathrm W^{1,p}( {\mathbb S}^1) $\end{document} and carré du champ methods

2020 ◽  
Vol 40 (1) ◽  
pp. 375-394
Author(s):  
Jean Dolbeault ◽  
◽  
Marta García-Huidobro ◽  
Rául Manásevich ◽  
◽  
...  
Keyword(s):  
Interpolation Inequalities ◽  
Carré Du Champ

scholarly journals Extremal functions for Caffarelli—Kohn—Nirenberg and logarithmic Hardy inequalities

2012 ◽  
Vol 142 (4) ◽  
pp. 745-767 ◽  
Author(s):  
Jean Dolbeault ◽  
Maria J. Esteban
Keyword(s):  
Sobolev Inequality ◽  
Existence Result ◽  
Logarithmic Sobolev Inequality ◽  
Extremal Functions ◽  
General Criterion ◽  
Limit Case ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Optimal Constants

We consider a family of Caffarelli–Kohn–Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities that were obtained recently as a limit case of the Caffarelli–Kohn–Nirenberg inequalities. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo–Nirenberg interpolation inequalities and Gross's logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.

Symmetry breaking in convergent-beam diffraction

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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