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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Korn inequality canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801476 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Korn inequality Context triple: [Poincaré inequality, relatedTo, Korn inequality]
-
A.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
B.
Hardy inequality
The Hardy inequality is a fundamental result in mathematical analysis that provides bounds on integrals or sums involving a function and its distance from a point, with important applications in functional analysis and partial differential equations.
-
C.
Fefferman–Phong inequality
The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
-
D.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
E.
Gagliardo–Nirenberg interpolation inequalities
The Gagliardo–Nirenberg interpolation inequalities are fundamental results in functional analysis and partial differential equations that bound intermediate norms of functions by combinations of lower and higher order norms, playing a key role in regularity theory and nonlinear analysis.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Korn inequality Target entity description: 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.
-
A.
Poincaré inequality
The Poincaré inequality is a fundamental result in functional analysis and partial differential equations that bounds the average oscillation of a function by the size of its gradient, playing a key role in Sobolev space theory and the study of elliptic problems.
-
B.
Hardy inequality
The Hardy inequality is a fundamental result in mathematical analysis that provides bounds on integrals or sums involving a function and its distance from a point, with important applications in functional analysis and partial differential equations.
-
C.
Fefferman–Phong inequality
The Fefferman–Phong inequality is a fundamental result in harmonic analysis and partial differential equations that provides weighted \(L^2\) estimates controlling functions by their gradients and associated potentials.
-
D.
Minkowski inequality
The Minkowski inequality is a fundamental result in functional analysis and measure theory that generalizes the triangle inequality to L^p spaces, providing a key tool for studying norms and integrable functions.
-
E.
Gagliardo–Nirenberg interpolation inequalities
The Gagliardo–Nirenberg interpolation inequalities are fundamental results in functional analysis and partial differential equations that bound intermediate norms of functions by combinations of lower and higher order norms, playing a key role in regularity theory and nonlinear analysis.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Korn inequality Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.