Gagliardo–Nirenberg interpolation inequality

In mathematics, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the ${\displaystyle L^{p}}$ norms of the weak derivatives of a function. 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[]

Suppose ${\displaystyle n,j,m}$ are non-negative integers and that ${\displaystyle 1\leq p,q,r\leq \infty }$ and ${\displaystyle \alpha \in [0,1]}$ are real numbers such that

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

and

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

Suppose furthermore that ${\displaystyle u:\mathbb {R} ^{n}\to \mathbb {R} }$ is a function in ${\displaystyle L^{q}(\mathbb {R} ^{n})}$ with m^th weak derivative in ${\displaystyle L^{r}(\mathbb {R} ^{n})}$.

Then the j^th weak derivative of ${\displaystyle u}$ lies in ${\displaystyle L^{p}(\mathbb {R} ^{n})}$ and there exists a constant ${\displaystyle C}$ depending on ${\displaystyle m,n,j,q,r}$ and ${\displaystyle \alpha }$ but independent of ${\displaystyle u}$ 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 ${\displaystyle j=0,mr<n}$ and ${\displaystyle q=\infty }$, then it is necessary to make the additional assumption that either ${\displaystyle \lim _{|x|\to \infty }u(x)=0}$ or that ${\displaystyle u\in L^{s}(\mathbb {R} ^{n})}$ L^s for some ${\displaystyle s<\infty }$
2. If ${\displaystyle 1<r<\infty }$ and ${\textstyle m-j-{\frac {n}{r}}}$ is a non-negative integer, then it is necessary to assume also that ${\displaystyle \alpha \neq 1}$

For functions ${\displaystyle u:\Omega \to \mathbb {R} }$ defined on a bounded Lipschitz domain ${\displaystyle \Omega \subseteq \mathbb {R} ^{n}}$, the interpolation inequality has the same hypotheses as above and reads

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

for arbitrary ${\displaystyle s\geq 1}$, where the constants ${\displaystyle C_{1},C_{2}}$ depend on the domain ${\displaystyle \Omega }$ and on ${\displaystyle s}$ in addition to the other parameters.

Consequences[]

• When ${\displaystyle \alpha =1}$, the term ${\displaystyle \|u\|_{L^{q}}}$vanishes 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 ${\displaystyle m=1,j=0,q=r=2,p=4}$ and ${\displaystyle n}$ is either ${\displaystyle 2,3}$. For example, the Ladyzheskaya inequality for dimension ${\displaystyle 2}$ states that ${\textstyle \|u\|_{L^{4}}\leq \|u\|_{L^{2}}^{1/2}\|Du\|_{L^{2}}^{1/2}}$
• 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.^[1]

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