Sobolev inequality
E620673
The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
All labels observed (7)
| Label | Occurrences |
|---|---|
| Sobolev inequalities | 4 |
| Sobolev embedding theorem | 2 |
| Sobolev embeddings | 2 |
| Sobolev inequality canonical | 2 |
| Sobolev inequality on bounded domains | 1 |
| Sobolev inequality on manifolds | 1 |
| Sobolev inequality with boundary conditions | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801474 — 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: Sobolev inequality Context triple: [Poincaré inequality, relatedTo, Sobolev 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.
Sobolev spaces
Sobolev spaces are function spaces that incorporate both functions and their weak derivatives, providing a fundamental framework for studying partial differential equations and variational problems.
-
C.
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.
-
D.
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.
-
E.
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.
- 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: Sobolev inequality Target entity description: The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
-
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.
Sobolev spaces
Sobolev spaces are function spaces that incorporate both functions and their weak derivatives, providing a fundamental framework for studying partial differential equations and variational problems.
-
C.
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.
-
D.
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.
-
E.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical inequality
ⓘ
result in functional analysis ⓘ result in partial differential equations ⓘ |
| appliesTo |
Sobolev space W^{k,p}(\Omega)
ⓘ
domains in R^n ⓘ functions with weak derivatives ⓘ |
| assumption | often assumes suitable regularity or geometry of the domain ⓘ |
| condition | usually assumes 1 \le p < n for first-order inequalities on R^n ⓘ |
| constantType | best constant often related to extremal functions ⓘ |
| coreStatement | bounds the L^q norm of a function by the L^p norm of its derivatives under suitable conditions ⓘ |
| defines | Sobolev conjugate exponent p^* = np/(n-p) ⓘ |
| dimensionDependence | constants depend on the space dimension n ⓘ |
| field |
functional analysis
ⓘ
mathematical analysis ⓘ partial differential equations ⓘ |
| generalizationOf | classical embedding of C_c^1 functions into L^q spaces ⓘ |
| hasVariant |
Sobolev inequality on bounded domains
ⓘ
linked to:
Sobolev inequality
Sobolev inequality on manifolds ⓘ
linked to:
Sobolev inequality
Sobolev inequality with boundary conditions ⓘ
linked to:
Sobolev inequality
critical Sobolev inequality ⓘ fractional Sobolev inequality ⓘ subcritical Sobolev inequality ⓘ weighted Sobolev inequality ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| implies |
compact embedding on bounded domains under additional conditions
ⓘ
continuous embedding of W^{1,p}(\mathbb{R}^n) into L^{p^*}(\mathbb{R}^n) ⓘ |
| importance |
central in the theory of Sobolev spaces
ⓘ
fundamental tool in modern PDE theory ⓘ |
| namedAfter | Sergei Sobolev ⓘ |
| relatedConcept |
Morrey inequality
ⓘ
Rellich–Kondrachov compactness theorem ⓘ |
| relatesTo |
Gagliardo–Nirenberg inequality
ⓘ
L^p spaces ⓘ Lebesgue spaces ⓘ Poincaré inequality ⓘ Sobolev embeddings ⓘ
linked to:
Sobolev inequality
Sobolev spaces ⓘ elliptic partial differential equations ⓘ interpolation inequalities ⓘ isoperimetric inequality ⓘ regularity theory for PDEs ⓘ weak derivatives ⓘ |
| type |
a priori estimate
ⓘ
embedding inequality ⓘ |
| typicalForm | \|u\|_{L^{p^*}(\mathbb{R}^n)} \le C \|\nabla u\|_{L^p(\mathbb{R}^n)} for 1 \le p < n and p^* = np/(n-p) ⓘ |
| usedFor |
controlling nonlinear terms in PDEs
ⓘ
energy estimates ⓘ establishing regularity of weak solutions ⓘ proving existence of weak solutions to PDEs ⓘ variational methods in calculus of variations ⓘ |
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: Sobolev inequality Description of subject: The Sobolev inequality is a fundamental result in functional analysis and partial differential equations that bounds the size of a function in certain Lebesgue spaces by the size of its derivatives, enabling key embedding and regularity properties.
Referenced by (13)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality
linked to: Sobolev inequality