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Gagliardo-Nirenberg interpolation inequality

In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the $L^{p}$-norms of different weak derivatives of a function. The theorem is of particular importance in the framework of elliptic partial differential equations. It was originally proposed by Emilio Gagliardo and Louis Nirenberg in two independent contributions during the International Congress of Mathematicians held in Edinburgh from August 14, 1958 through August 21, 1958. In the following year, both authors improved their results and published them independently.

Statement of the inequality

For any extended real (i.e. possibly infinite) positive quantity $p\geq 1$ and any integer $k\geq 1$, let $L^{p}(\mathbb {R} ^{n})$ denote the usual $L^{p}$ spaces, while $W^{k,p}(\mathbb {R} ^{n})$ denotes the Sobolev space consisting of all real-valued functions in $L^{p}(\mathbb {R} ^{n})$ such that all their weak derivatives up to order $k$ are also in $L^{p}(\mathbb {R} ^{n})$. Both families of spaces are intended to be endowed with their standard norms, namely:$$\|u\|_{L^{p}(\mathbb {R} ^{n})}:={\begin{cases}\left(\displaystyle \int _{\mathbb {R} ^{n}}|u|^{p}\right)^{\frac {1}{p}}&\quad {\text{if }}p<+\infty ,\\{\underset {\mathbb {R} ^{n}}{\operatorname {ess\,sup} }}\,|u|&\quad {\text{if }}p=+\infty ;\end{cases}}\qquad \qquad \|u\|_{W^{k,p}(\mathbb {R} ^{n})}:={\begin{cases}\left(\|u\|_{L^{p}(\mathbb {R} ^{n})}^{p}+\displaystyle \sum _{\alpha =1}^{k}\|D^{\alpha }u\|_{L^{p}(\mathbb {R} ^{n})}^{p}\right)^{\frac {1}{p}}&\quad {\text{if }}p<+\infty ,\\\|u\|_{L^{\infty }(\mathbb {R} ^{n})}+\displaystyle \sum _{\alpha =1}^{k}\|D^{\alpha }u\|_{L^{\infty }(\mathbb {R} ^{n})}&\quad {\text{if }}p=+\infty .\end{cases}}$$Above, for the sake of convenience, the same notation is used for scalar, vector and tensor-valued Lebesgue and Sobolev spaces.

The original version of the theorem, for functions defined on the whole Euclidean space $\mathbb {R} ^{n}$, can be stated as follows.

Theorem (Gagliardo-Nirenberg) — Let $1\leq q\leq +\infty$ be a positive extended real quantity. Let $j$ and $m$ be non-negative integers such that $j<m$. Furthermore, let $1\leq p,\,r\leq +\infty$ be two positive extended real quantities and $\theta \in [0,1]$ such that the inequalities $${\dfrac {1}{p}}={\dfrac {j}{n}}+\theta \left({\dfrac {1}{r}}-{\dfrac {m}{n}}\right)+{\dfrac {1-\theta }{q}},\qquad {\dfrac {j}{m}}\leq \theta \leq 1$$ hold. Then, $$\|D^{j}u\|_{L^{p}(\mathbb {R} ^{n})}\leq C\|D^{m}u\|_{L^{r}(\mathbb {R} ^{n})}^{\theta }\|u\|_{L^{q}(\mathbb {R} ^{n})}^{1-\theta }$$ for any $u\in L^{q}(\mathbb {R} ^{n})$ such that $D^{m}u\in L^{r}(\mathbb {R} ^{n})$, with two exceptional cases:

1. if $j=0$ (with the understanding that $D^{0}u=u$), $q=+\infty$ and $rm<n$, then an additional assumption is needed: either $u$ tends to 0 at infinity, or $u\in L^{s}(\mathbb {R} ^{n})$ for some finite value of $s$;
2. if $r>1$ and $m-j-{\frac {n}{r}}$ is a non-negative integer, then the additional assumption ${\frac {j}{m}}\leq \theta <1$ (notice the strict inequality) is needed.

In any case, the constant $C>0$ depends on the parameters $j,\,m,\,n,\,q,\,r,\,\theta$, but not on $u$.

Relevant corollaries of the Gagliardo-Nirenberg inequality

The Gagliardo-Nirenberg inequality generalizes a collection of well-known results in the field of functional analysis. Indeed, given a suitable choice of the seven parameters appearing in the statement of the theorem, one obtains several useful and recurring inequalities in the theory of partial differential equations:

• The Sobolev embedding theorem establishes the existence of continuous embeddings between Sobolev spaces with different orders of differentiation and/or integrability. It can be obtained from the Gagliardo-Nirenberg inequality setting $\theta =1$ (so that the choice of $q$ becomes irrelevant, and the same goes for the associated requirement $u\in L^{q}(\mathbb {R} ^{n})$) and the remaining parameters in such a way that $${\dfrac {1}{p}}-{\dfrac {j}{n}}={\dfrac {1}{r}}-{\dfrac {m}{n}}$$and the other hypotheses are satisfied. The result reads then $$\|D^{j}u\|_{L^{p}(\mathbb {R} ^{n})}\leq C\|D^{m}u\|_{L^{r}(\mathbb {R} ^{n})}$$for any $u$ such that $D^{m}u\in L^{r}(\mathbb {R} ^{n})$. In particular, setting $j=0$ and $m=1$ yields that $p=r^{*}$, namely the Sobolev conjugate exponent of $r$, and we have the embedding $$W^{1,r}(\mathbb {R} ^{n})\hookrightarrow L^{r^{*}}(\mathbb {R} ^{n}).$$Notice that, in the embedding above, we also implicitly assume that $u\in L^{r}(\mathbb {R} ^{n})$ and hence the first exceptional case does not apply.
• The Ladyzhenskaya inequality is a special case of the Gagliardo-Nirenberg inequality in both cases $n=2$ and $n=3.$ Indeed, the former case corresponds to the parameter choice $$j=0,\quad m=1,\quad p=4,\quad q=r=2,\quad \theta ={\frac {1}{2}},$$yielding $$\|u\|_{L^{4}(\mathbb {R} ^{2})}\leq C\|u\|_{L^{2}(\mathbb {R} ^{2})}^{\frac {1}{2}}\|\nabla u\|_{L^{2}(\mathbb {R} ^{2})}^{\frac {1}{2}}$$for any $u\in W^{1,2}(\mathbb {R} ^{2}).$ The constant $C$ is universal and can be proven to be $2^{-{\frac {1}{4}}}$. In three space dimensions a slightly different choice of parameters is needed, namely$$j=0,\quad m=1,\quad p=4,\quad q=r=2,\quad \theta ={\frac {3}{4}},$$yielding$$\|u\|_{L^{4}(\mathbb {R} ^{3})}\leq C\|u\|_{L^{2}(\mathbb {R} ^{3})}^{\frac {1}{4}}\|\nabla u\|_{L^{2}(\mathbb {R} ^{3})}^{\frac {3}{4}}$$for any $u\in W^{1,2}(\mathbb {R} ^{3})$. Here, it holds $C=\left({\frac {4{\sqrt {3}}}{9}}\right)^{\frac {3}{4}}$.
• The Agmon inequality is also a special case of the Gagliardo-Nirenberg inequality. When $n=2$, the correct parameter choice is $$j=0,\quad m=2,\quad p=+\infty ,\quad q=r=2,\quad \theta ={\frac {1}{2}},$$yielding $$\|u\|_{L^{\infty }(\mathbb {R} ^{2})}\leq C\|u\|_{L^{2}(\mathbb {R} ^{2})}^{\frac {1}{2}}\|D^{2}u\|_{L^{2}(\mathbb {R} ^{2})}^{\frac {1}{2}}$$for any $u\in L^{2}(\mathbb {R} ^{2})$ such that $D^{2}u\in L^{2}(\mathbb {R} ^{2}).$ In three spatial dimensions, the correct inequality corresponds to $$j=0,\quad m=2,\quad p=+\infty ,\quad q=6,\quad r=2,\quad \theta ={\frac {1}{2}},$$and thus$$\|u\|_{L^{\infty }(\mathbb {R} ^{3})}\leq C\|u\|_{L^{6}(\mathbb {R} ^{3})}^{\frac {1}{2}}\|D^{2}u\|_{L^{2}(\mathbb {R} ^{3})}^{\frac {1}{2}}.$$
• The Nash inequality, which was published by John Nash in 1958, is yet another result generalized by the Gagliardo-Nirenberg inequality. Indeed, choosing$$j=0,\quad m=1,\quad n\geq 1,\quad p=2,\quad q=1,\quad r=2,\quad \theta ={\frac {n}{n+2}},$$one gets $$\|u\|_{L^{2}(\mathbb {R} ^{n})}\leq C\|u\|_{L^{1}(\mathbb {R} ^{n})}^{\frac {2}{n+2}}\|\nabla u\|_{L^{2}(\mathbb {R^{n}} )}^{\frac {n}{n+2}}$$which is oftentimes recast as$$\|u\|_{L^{2}(\mathbb {R} ^{n})}^{1+{\frac {2}{n}}}\leq C\|u\|_{L^{1}(\mathbb {R} ^{n})}^{\frac {2}{n}}\|\nabla u\|_{L^{2}(\mathbb {R^{n}} )}$$or its squared version.

The Gagliardo-Nirenberg inequality in bounded domains

In many problems coming from the theory of partial differential equations, and especially in the ones linked with real-life applications, one has to deal with functions whose domain is not the whole Euclidean space $\mathbb {R} ^{n}$, but rather some given bounded, open and connected set $\Omega \subset \mathbb {R} ^{n}.$ In the following, we also assume that $\Omega$ has finite Lebesgue measure and a Lipschitz boundary. Both Gagliardo and Nirenberg found out that their theorem could be extended to this case adding a penalization term to the right hand side. Precisely,

Theorem (Gagliardo-Nirenberg in bounded domains) — Let $\Omega \subset \mathbb {R} ^{n}$ be a measurable, bounded, open and connected domain with Lipschitz boundary. Let $1\leq q\leq +\infty$ be a positive extended real quantity. Let $j$ and $m$ be non-negative integers such that $j<m$. Furthermore, let $1\leq p,\,r\leq +\infty$ be two positive extended real quantities and $\theta \in [0,1]$ such that the inequalities $${\dfrac {1}{p}}={\dfrac {j}{n}}+\theta \left({\dfrac {1}{r}}-{\dfrac {m}{n}}\right)+{\dfrac {1-\theta }{q}},\qquad {\dfrac {j}{m}}\leq \theta \leq 1$$ hold. Then, $$\|D^{j}u\|_{L^{p}(\Omega )}\leq C\|D^{m}u\|_{L^{r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }+C\|u\|_{L^{\sigma }(\Omega )}$$ where $u\in L^{q}(\Omega )$ such that $D^{m}u\in L^{r}(\Omega )$ and $\sigma$ is arbitrary, with one exceptional case:

1. if $r>1$ and $m-j-{\frac {n}{r}}$ is a non-negative integer, then the additional assumption ${\frac {j}{m}}\leq \theta <1$ (notice the strict inequality) is needed.

In any case, the constant $C>0$ depends on the parameters $j,\,m,\,n,\,q,\,r,\,\theta$, on the domain $\Omega$, but not on $u$.

The necessity of a different formulation with respect to the case $\Omega =\mathbb {R} ^{n}$ is rather straightforward to prove. Indeed, since $\Omega$ has finite Lebesgue measure, any constant function belongs to $L^{p}(\Omega )$ for every $p$ (including $p=+\infty$). Of course, it holds much more: constant functions belong to $C^{\infty }(\Omega )$ and all their derivatives are identically equal to zero in $\Omega$. It can be easily seen that the Gagliardo-Nirenberg inequality for the case $\Omega =\mathbb {R} ^{n}$ fails to be true for any nonzero constant function, since a contradiction is immediately achieved, and therefore cannot hold in general for integrable functions defined on bounded domains.

That being said, under slightly stronger assumptions, it is possible to recast the theorem in such a way that the penalization term is "absorbed" in the first term at right hand side. Indeed, if $u\in L^{q}(\Omega )\cap W^{m,r}(\Omega )$, then one can choose $\sigma =\min(q,r)$ and get $${\begin{aligned}\|D^{j}u\|_{L^{p}(\Omega )}&\leq C\|D^{m}u\|_{L^{r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }+C\|u\|_{L^{\min(r,q)}(\Omega )}^{\theta }\|u\|_{L^{\min(r,q)}(\Omega )}^{1-\theta }\\&\leq C\|u\|_{W^{m,r}(\Omega )}^{\theta }\|u\|_{L^{q}(\Omega )}^{1-\theta }.\end{aligned}}$$This formulation has the advantage of recovering the structure of the theorem in the full Euclidean space, with the only caution that the Sobolev seminorm is replaced by the full $W^{m,r}$-norm. For this reason, the Gagliardo-Nirenberg inequality in bounded domains is commonly stated in this way.