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 L^p 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

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

and

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

Then

1. every function u: Rⁿ → R that lies in L^q(Rⁿ) with m^th derivative in L^r(Rⁿ) also has j^th derivative in L^p(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 L^s 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 L^q 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.
• In the setting of the Sobolev spaces ${\displaystyle H^{s}}$, with ${\displaystyle 0\leq s_{1}<s<s_{2}}$, a special case is given by ${\displaystyle \|u\|_{H^{s}}\leq \|u\|_{H^{s_{1}}}^{\frac {s_{2}-s}{s_{2}-s_{1}}}\|u\|_{H^{s_{2}}}^{\frac {s-s_{1}}{s_{2}-s_{1}}}}$. This can also be derived via Plancherel theorem and Hölder's inequality.

References

• E. Gagliardo. Ulteriori proprietà di alcune classi di funzioni in più variabili. Ricerche Mat.,8:24–51, 1959.
• Nirenberg, L. (1959). "On elliptic partial differential equations". Ann. Scuola Norm. Sup. Pisa (3). 13: 115–162.
• Haïm Brezis, Petru Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Annales de l’Institut Henri Poincaré - Non Linear Analysis 35 (2018), 1355-1376.
• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8
• Nguyen-Anh Dao, Jesus Ildefonso Diaz, Quoc-Hung Nguyen (2018), Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces and BMO, Nonlinear Analysis, Volume 173, Pages 146-153.