Gagliardo–Nirenberg interpolation inequalities

E588689

mathematical inequality result in functional analysis tool in partial differential equations

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.

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Observed surface forms (1)

Surface form	Occurrences
Gagliardo–Nirenberg interpolation inequality	1

Statements (49)

Predicate	Object
instanceOf	mathematical inequality ⓘ result in functional analysis ⓘ tool in partial differential equations ⓘ
appliesTo	Sobolev functions ⓘ functions on Euclidean space ⓘ functions on domains in R^n ⓘ vector-valued functions ⓘ
condition	0 ≤ a ≤ 1 ⓘ 0 ≤ j < m ⓘ 1 ≤ p,q,r ≤ ∞ ⓘ scaling relation 1/r - j/n = a(1/p - m/n) + (1-a)/q ⓘ
field	Sobolev space theory ⓘ functional analysis ⓘ nonlinear analysis ⓘ partial differential equations ⓘ
hasForm	\|\|D^j u\|\|_{L^r} ≤ C \|\|D^m u\|\|_{L^p}^a \|\|u\|\|_{L^q}^{1-a} ⓘ
implies	compactness properties in Sobolev embeddings ⓘ
namedAfter	Emilio Gagliardo NERFINISHED ⓘ Louis Nirenberg NERFINISHED ⓘ
purpose	to bound intermediate norms by lower and higher order norms ⓘ to control nonlinear terms in PDEs ⓘ to interpolate between different Sobolev norms ⓘ to obtain regularity estimates for solutions of PDEs ⓘ
relatesConcept	H^s spaces ⓘ L^p norms ⓘ L^q norms ⓘ Lebesgue spaces NERFINISHED ⓘ Navier–Stokes equations NERFINISHED ⓘ Sobolev embeddings NERFINISHED ⓘ Sobolev inequalities NERFINISHED ⓘ Sobolev norms NERFINISHED ⓘ W^{k,p} spaces ⓘ a priori estimates ⓘ critical exponents ⓘ derivative estimates ⓘ elliptic equations ⓘ intermediate norms ⓘ interpolation theory ⓘ nonlinear PDEs ⓘ parabolic equations ⓘ quasilinear PDEs ⓘ reaction–diffusion equations ⓘ regularity theory ⓘ scaling arguments ⓘ semilinear PDEs ⓘ
usedFor	blow-up criteria in nonlinear PDEs ⓘ bootstrap arguments in regularity theory ⓘ energy estimates in evolution equations ⓘ global existence proofs for PDEs ⓘ

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

Louis Nirenberg → knownFor → Gagliardo–Nirenberg interpolation inequalities ⓘ

Sobolev spaces → relatedTheorem → Gagliardo–Nirenberg interpolation inequalities ⓘ

this entity surface form: Gagliardo–Nirenberg interpolation inequality