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 -norms of different weak derivatives of a function through an interpolation inequality. 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 Haim Brezis and Petru Mironescu in the late 2010s.
History
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. In the following year, both authors improved their results and published them independently. 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. For example, Nirenberg firstly included the inequality in a collection of lectures given in Pisa from September 1 to September 10, 1958. The transcription of the lectures was later published in 1959, and the author explicitly states only the main steps of the proof. 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. A detailed proof in the whole Euclidean space was published in 2021.
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, which was addressed and refined by Nirenberg later in 1966. The investigation of Gagliardo-Nirenberg type inequalities continued in the following decades. For instance, a careful study on negative exponents has been carried out extending the work of Nirenberg in 2018, while Brezis and Mironescu characterized in full generality the embeddings between Sobolev spaces extending the inequality to fractional orders.
Statement of the inequality
For any extended real (i.e. possibly infinite) positive quantity and any integer , let denote the usual spaces, while denotes the Sobolev space consisting of all real-valued functions in such that all their weak derivatives up to order are also in . Both families of spaces are intended to be endowed with their standard norms, namely: where stands for essential supremum. 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 , can be stated as follows.
Notice that the parameter is determined uniquely by all the other ones and usually assumed to be finite. However, there are sharper formulations in which is considered (but other values may be excluded, for example ).
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:
