Approximation of the operator of differentiation in a weighted Sobolev class

1991 ◽  
Vol 49 (5) ◽  
pp. 550-551
Author(s):  
K. T. Mynbaev
Keyword(s):  
Sobolev Class ◽  
Weighted Sobolev Class

scholarly journals On the Stability in Bonnet's Theorem of the Surface Theory

2007 ◽  
Vol 14 (3) ◽  
pp. 543-564
Author(s):  
Yuri G. Reshetnyak
Keyword(s):  
Differential Operators ◽  
Sobolev Class ◽  
Integrability Condition ◽  
Complete Integrability ◽  
Integral Representations ◽  
Completely Integrable Systems ◽  
Frobenius Theorem ◽  
Fundamental Tensor ◽  
The Stability ◽  
Codazzi Equations

Abstract In the space , 𝑛-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class with 𝑝 > 𝑛. The first and the second fundamental tensor are defined. The Peterson–Codazzi equations for such functions are understood in some generalized sense. It is proved that if the first and the second fundamental tensor of one surface are close to the first and, respectively, to the second fundamental tensor of the other surface, then these surfaces will be close up to the motion of the space . A difference between the fundamental tensors and the nearness of the surfaces are measured with the help of suitable 𝑊-norms. The proofs are based on a generalization of Frobenius' theorem about completely integrable systems of the differential equations which was proved by Yu. E. Borovskiĭ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.

Sign in / Sign up

Export Citation Format

Share Document