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.
Statement of the inequality
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
- every function u: Rn → R that lies in Lq(Rn) with mth derivative in Lr(Rn) also has jth derivative in Lp(Rn);
- and, furthermore, there exists a constant C depending only on m, n, j, q, r and α such that
The result has two exceptional cases:
- If j = 0, mr < n and q = ∞, then it is necessary to make the additional assumption that either u tends to zero at infinity or that u lies in Ls for some finite s > 0.
- If 1 < r < ∞ and m − j − n ⁄ r is a non-negative integer, then it is necessary to assume also that α ≠ 1.
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.
- When α = 1, the Lq norm of u vanishes from the inequality, and the Gagliardo–Nirenberg interpolation inequality then implies the Sobolev embedding theorem. (Note, in particular, that r is permitted to be 1.)
- Another special case of the Gagliardo–Nirenberg interpolation inequality is Ladyzhenskaya's inequality, in which m = 1, j = 0, n = 2 or 3, q and r are both 2, and p = 4.