TheInfoList

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 uRn → 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

${\displaystyle {\frac {1}{p}}={\frac {j}{n}}+\left({\frac {1}{r}}-{\frac {m}{n}}\right)\alpha +{\frac {1-\alpha }{q}}}$

and

${\displaystyle {\frac {j}{m}}\leq \alpha \leq 1.}$

Then

1. every function uRn → R that lies in Lq(Rn) with mth derivative in Lr(Rn) also has jth derivative in Lp(Rn);
2. and, furthermore, there exists a constant C depending only on m, n, j, q, r and α such that
${\displaystyle \\mathrm {D} ^{j}u\_{L^{p}}\leq C\\mathrm {D} ^{m}u\_{L^{r}}^{\alpha }\u\_{L^{q}}^{1-\alpha }.}$

The result has two exceptional cases:

1. 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.
2. 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

${\displaystyle \\mathrm {D} ^{j}u\_{L^{p}}\leq C_{1}\\mathrm {D} ^{m}u\_{L^{r}}^{\alpha }\u\_{L^{q}}^{1-\alpha }+C_{2}\u\_{L^{s}}}$

where s > 0 is arbitrary; naturally, the constants C1 and C2 depend upon the domain Ω as well as m, n etc.

## Consequences

• 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.