On the Stability in Bonnet's Theorem of the Surface Theory
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Yuri G. Reshetnyak
Abstract
In the space 
, š-dimensional surfaces are considered having the parametrizations which are functions of the Sobolev class 
with š > š. The first and the second fundamental tensor are defined. The PetersonāCodazzi 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 š-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. BorovskiÄ. The integral representations of functions by differential operators with complete integrability condition are used, which were elaborated by the author in his other works.
Ā© Heldermann Verlag
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- The Robin Function and Its Eigenvalues
- Finite Dimensional Grading of the Virasoro Algebra
- On Nonmeasurable Subgroups of Uncountable Solvable Groups
- On a Priori Estimates of Solutions of Systems of Higher Order Nonlinear Functional-Differential Inequalities
- On Doubly Periodic Solutions of Nonlinear Hyperbolic Equations of Higher Order
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- On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak
- Stagnation Zones of š“-Solutions
- On a Periodic Boundary Value Problem for Fourth Order Linear Functional Differential Equations
- On the Stability in Bonnet's Theorem of the Surface Theory
- On Some Properties of Solutions of Polyharmonic Equation in Polyhedral Angles
- Generalized Analytic Functions in Higher Dimensions
Articles in the same Issue
- The Robin Function and Its Eigenvalues
- Finite Dimensional Grading of the Virasoro Algebra
- On Nonmeasurable Subgroups of Uncountable Solvable Groups
- On a Priori Estimates of Solutions of Systems of Higher Order Nonlinear Functional-Differential Inequalities
- On Doubly Periodic Solutions of Nonlinear Hyperbolic Equations of Higher Order
- Analytic Function Theory in Algebras
- Complex Geometry of the Universal TeichmĆ¼ller Space. II
- On Solvability of the Neumann Problem in an Energy Space for a Domain with Peak
- Stagnation Zones of š“-Solutions
- On a Periodic Boundary Value Problem for Fourth Order Linear Functional Differential Equations
- On the Stability in Bonnet's Theorem of the Surface Theory
- On Some Properties of Solutions of Polyharmonic Equation in Polyhedral Angles
- Generalized Analytic Functions in Higher Dimensions
