Sobolev inequality

E620673

mathematical inequality result in functional analysis result in partial differential equations

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.

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Observed surface forms (2)

Surface form	Occurrences
Sobolev embedding theorem	1
Sobolev inequalities	1

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 NERFINISHED ⓘ Sobolev inequality on manifolds NERFINISHED ⓘ Sobolev inequality with boundary conditions NERFINISHED ⓘ 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 NERFINISHED ⓘ
relatedConcept	Morrey inequality NERFINISHED ⓘ Rellich–Kondrachov compactness theorem NERFINISHED ⓘ
relatesTo	Gagliardo–Nirenberg inequality NERFINISHED ⓘ L^p spaces ⓘ Lebesgue spaces NERFINISHED ⓘ Poincaré inequality NERFINISHED ⓘ Sobolev embeddings NERFINISHED ⓘ Sobolev spaces NERFINISHED ⓘ 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 ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Poincaré inequality → relatedTo → Sobolev inequality ⓘ

Young's inequality → hasApplication → Sobolev inequality ⓘ

this entity surface form: Sobolev inequalities

Sobolev spaces → relatedTheorem → Sobolev inequality ⓘ

this entity surface form: Sobolev embedding theorem