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Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein 1970, Chapter V, §1.3). Let 0 < α < n and 1 < p < q < ∞.
In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the -norms of different weak derivatives of a function through an interpolation inequality.
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations , including the theory of harmonic maps .
When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality. The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have ...
The Sobolev spaces , for < are defined as the closure of the set of compactly supported test functions with respect to the , ()-norm. The following alternative characterization holds: The following alternative characterization holds:
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In mathematics, Friedrichs's inequality is a theorem of functional analysis, due to Kurt Friedrichs. It places a bound on the L p norm of a function using L p bounds on the weak derivatives of the function and the geometry of the domain , and can be used to show that certain norms on Sobolev spaces are equivalent.
(However, today the customary name is "compactness theorem", whereas "selection theorem" has a precise and quite different meaning, referring to set-valued functions.) The Rellich–Kondrachov theorem may be used to prove the Poincaré inequality , [ 2 ] which states that for u ∈ W 1, p (Ω; R ) (where Ω satisfies the same hypotheses as above),