“Korn's inequality”的意思、由来-开放百科全书

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Korn's inequality

释义

Statement of the inequality
See also
References
External links

In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric at every point, then the gradient must be equal to a constant skew-symmetric matrix. Korn's theorem is a quantitative version of this statement, which intuitively says that if the gradient of a vector field is on average not far from the space of skew-symmetric matrices, then the gradient must not be far from a particular skew-symmetric matrix. The statement that Korn's inequality generalizes thus arises as a special case of rigidity.

In (linear) elasticity theory, the symmetric part of the gradient is a measure of the strain that an elastic body experiences when it is deformed by a given vector-valued function. The inequality is therefore an important tool as an a priori estimate in linear elasticity theory.

Statement of the inequality

Let {{math|Ω}} be an open, connected domain in {{math|n}}-dimensional Euclidean space {{math|Rⁿ}}, {{math|n ≥ 2}}. Let {{math|H¹(Ω)}} be the Sobolev space of all vector fields {{math|v = (v¹, ..., vⁿ)}} on {{math|Ω}} that, along with their (first) weak derivatives, lie in the Lebesgue space {{math|L²(Ω)}}. Denoting the partial derivative with respect to the i^th component by {{math|∂_i}}, the norm in {{math|H¹(Ω)}} is given by

Then there is a constant {{math|C ≥ 0}}, known as the Korn constant of {{math|Ω}}, such that, for all {{math|v ∈ H¹(Ω)}},

{{NumBlk|:|

|{{EquationRef|1}}}}

where {{math|e}} denotes the symmetrized gradient given by

Inequality {{EquationNote|1|(1)}} is known as Korn's inequality.

References

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External links

{{SpringerEOM

| title=Korn inequality
| id=Korn_inequality&oldid=30665
| last=Voitsekhovskii
| first=M. I.
| author-link=
}}

3 : Inequalities|Sobolev spaces|Solid mechanics

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Statement of the inequality

See also

References

External links