Korn inequality
E620674
Korn inequality is a fundamental result in functional analysis and the mathematical theory of elasticity that provides bounds relating the full gradient of a vector field to its symmetric part, ensuring control of deformations by their strains.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in functional analysis ⓘ result in mathematical elasticity ⓘ |
| appliesTo |
displacement fields in elasticity
ⓘ
vector-valued Sobolev functions ⓘ |
| dependsOn | geometric properties of the domain ⓘ |
| ensures |
coercivity of the elasticity bilinear form
ⓘ
control of deformations by their strains ⓘ well-posedness of linear elasticity problems ⓘ |
| field |
functional analysis
ⓘ
mathematical theory of elasticity ⓘ |
| generalizationOf | Poincaré inequality for vector fields with symmetric gradient ⓘ |
| guarantees | equivalence of norms involving gradient and symmetric gradient under suitable conditions ⓘ |
| hasParameter | Korn constant ⓘ |
| hasVersion |
Korn inequality on bounded domains
NERFINISHED
ⓘ
Korn inequality on manifolds NERFINISHED ⓘ Korn inequality on unbounded domains NERFINISHED ⓘ Korn inequality with partial boundary conditions NERFINISHED ⓘ first Korn inequality ⓘ second Korn inequality ⓘ |
| historicalPeriod | early 20th century ⓘ |
| holdsOn | sufficiently regular domains ⓘ |
| implies | control of rigid body motions via boundary conditions ⓘ |
| involves |
L2 norms
ⓘ
Sobolev norms ⓘ strain tensor ⓘ symmetric gradient ⓘ |
| isToolFor |
compactness arguments in elasticity
ⓘ
derivation of plate and shell models from 3D elasticity ⓘ rigidity estimates ⓘ |
| KornConstantDependsOn | dimension of the space ⓘ |
| KornConstantDependsOn |
shape of the domain
ⓘ
type of boundary conditions ⓘ |
| languageOfOriginalWork | German ⓘ |
| namedAfter | Arthur Korn NERFINISHED ⓘ |
| relates |
full gradient of a vector field
ⓘ
symmetric part of the gradient ⓘ |
| requires | appropriate boundary conditions ⓘ |
| type |
a priori estimate
ⓘ
coercivity estimate ⓘ |
| usedFor |
establishing stability of numerical schemes in elasticity
ⓘ
proving existence of weak solutions in elasticity ⓘ |
| usedIn |
calculus of variations
ⓘ
continuum mechanics ⓘ finite element analysis ⓘ linear elasticity ⓘ nonlinear elasticity ⓘ partial differential equations ⓘ |
Referenced by (1)
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