Small parameter in the theory of isometric imbeddings for two-dimensional Riemannian manifolds into euclidean spaces

1995 ◽  
Vol 74 (3) ◽  
pp. 1078-1116 ◽  
Author(s):  
E. G. Poznyak ◽  
E. V. Shikin
Keyword(s):  
Small Parameter ◽  
Riemannian Manifolds ◽  
Two Dimensional ◽  
Euclidean Spaces ◽  
Isometric Imbeddings

scholarly journals Inequalities involving Aharonov–Bohm magnetic potentials in dimensions 2 and 3

2020 ◽  
pp. 2150006
Author(s):  
Denis Bonheure ◽  
Jean Dolbeault ◽  
Maria J. Esteban ◽  
Ari Laptev ◽  
Michael Loss
Keyword(s):  
Magnetic Field ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Euclidean Spaces ◽  
Hardy Inequalities ◽  
Interpolation Inequalities ◽  
Nonlinear Interpolation ◽  
Magnetic Potentials ◽  
Aharonov Bohm

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions [Formula: see text] and [Formula: see text].

scholarly journals Programing Two-Dimensional Materials in Non-Euclidean Spaces

Chem ◽  
2020 ◽  
Vol 6 (4) ◽  
pp. 829-831
Author(s):  
Shanshan Wang ◽  
Jin Zhang
Keyword(s):  
Two Dimensional ◽  
Euclidean Spaces ◽  
Two Dimensional Materials

AVERAGING PRINCIPLE FOR QUASI-LINEAR PARABOLIC PDEs AND RELATED DIFFUSION PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150008 ◽  
Author(s):  
MARK FREIDLIN ◽  
LEONID KORALOV
Keyword(s):  
Small Parameter ◽  
Diffusion Processes ◽  
First Integral ◽  
Second Order ◽  
Averaging Principle ◽  
Two Dimensional ◽  
Parabolic Pdes ◽  
Dimensional Flow ◽  
Linear Perturbations ◽  
Two Dimensional Flow

Quasi-linear perturbations of a two-dimensional flow with a first integral and the corresponding parabolic PDEs with a small parameter at the second-order derivatives are considered in this paper.

Extension of the Prandtl–Batchelor theorem to three-dimensional flows slowly varying in one direction

2010 ◽  
Vol 654 ◽  
pp. 351-361 ◽  
Author(s):  
M. SANDOVAL ◽  
S. CHERNYSHENKO
Keyword(s):  
Flow Field ◽  
Small Parameter ◽  
Order Approximation ◽  
Three Dimensional ◽  
Inviscid Limit ◽  
Two Dimensional ◽  
Leading Order ◽  
Dimensional Flow ◽  
Two Dimensional Flow

According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

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