The theorem is of particular importance in the framework of elliptic partial differential equations and was originally formulated by Emilio Gagliardo and Louis Nirenberg in 1958.
The Gagliardo-Nirenberg inequality has found numerous applications in the investigation of nonlinear partial differential equations, and has been generalized to fractional Sobolev spaces by Haïm Brezis and Petru Mironescu in the late 2010s.
The Gagliardo-Nirenberg inequality 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.
[3][4][5] Nonetheless, a complete proof of the inequality went missing in the literature for a long time.
Indeed, to some extent, both original works of Gagliardo and Nirenberg do not contain a full and rigorous argument proving the result.
The transcription of the lectures was later published in 1959, and the author explicitly states only the main steps of the proof.
[5] On the other hand, the proof of Gagliardo did not yield the result in full generality, i.e. for all possible values of the parameters appearing in the statement.
[6] From its original formulation, several mathematicians worked on proving and generalizing Gagliardo-Nirenberg type inequalities.
The Italian mathematician Carlo Miranda developed a first generalization in 1963,[7] which was addressed and refined by Nirenberg later in 1966.
For instance, a careful study on negative exponents has been carried out extending the work of Nirenberg in 2018,[9] while Brezis and Mironescu characterized in full generality the embeddings between Sobolev spaces extending the inequality to fractional orders.
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
[9] 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: A complete and detailed proof of the Gagliardo-Nirenberg inequality has been missing in literature for a long time since its first statements.
Indeed, both original works of Gagliardo and Nirenberg lacked some details, or even presented only the main steps of the proof.
A double induction argument is applied to the couple of integers
As base case, we assume that the Gagliardo-Nirenberg inequality holds for
Since, by the first induction step, we can assume the Gagliardo-Nirenberg inequality holds with
In order to prove the base case, several technical lemmas are necessary, while the remaining values of
can be recovered by interpolation and a proof can be found, for instance, in the original work of Nirenberg.
[5] In many problems coming from the theory of partial differential equations, one has to deal with functions whose domain is not the whole Euclidean space
has finite Lebesgue measure and satisfies the cone condition (among those are the widely used Lipschitz domains).
Both Gagliardo and Nirenberg found out that their theorem could be extended to this case adding a penalization term to the right hand side.
be a measurable, bounded, open and connected domain satisfying the cone condition.
fails to be true for any non constant affine function, since a contradiction is immediately achieved when
, and therefore cannot hold in general for integrable functions defined on bounded domains.
For this reason, the Gagliardo-Nirenberg inequality in bounded domains is commonly stated in this way.
[18] Finally, observe that the first exceptional case appearing in the statement of the Gagliardo-Nirenberg inequality for the whole space is no longer relevant in bounded domains, since for finite measure sets we have that
The problem of interpolating different Sobolev spaces has been solved in full generality by Haïm Brezis and Petru Mironescu in two works dated 2018 and 2019.
A generalization of the Gagliardo-Nirenberg inequality to these spaces reads Theorem[20] (Brezis-Mironescu) — Let