Gagliardo–Nirenberg interpolation inequality
In mathematics, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that estimates the weak derivatives of a function. The estimates are in terms of Lp norms of the function and its derivatives, and the inequality “interpolates” among various values of p and orders of differentiation, hence the name. The result is of particular importance in the theory of elliptic partial differential equations. It was proposed by Louis Nirenberg and Emilio Gagliardo.
The inequality concerns functions u: Rn → R. Fix 1 ≤q, r ≤ ∞ and a natural number m. Suppose also that a real number α and a natural number j are such that
and
Then
The result has two exceptional cases:
For functions u: Ω → R defined on a bounded Lipschitz domain Ω ⊆ Rn, the interpolation inequality has the same hypotheses as above and reads
where s > 0 is arbitrary; naturally, the constants C1 and C2 depend upon the domain Ω as well as m, n etc.
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