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 -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 edit

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.[1][2] In the following year, both authors improved their results and published them independently.[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. 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.[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] A detailed proof in the whole Euclidean space was published in 2021.[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.[8] 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,[9] while Brezis and Mironescu characterized in full generality the embeddings between Sobolev spaces extending the inequality to fractional orders.[10][11]

Statement of the inequality edit

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:[12]

 
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.

Theorem[13] (Gagliardo-Nirenberg) — Let   be a positive extended real quantity. Let   and   be non-negative integers such that  . Furthermore, let   be a positive extended real quantity,   be real and   such that the relations

 
hold. Then,
 
for any   such that  , with two exceptional cases:
  1. if   (with the understanding that  ),   and  , then an additional assumption is needed: either   tends to 0 at infinity, or   for some finite value of  ;
  2. if   and   is a non-negative integer, then the additional assumption   (notice the strict inequality) is needed.

In any case, the constant   depends on the parameters  , but not on  .

Notice that the parameter   is determined uniquely by all the other ones and usually assumed to be finite.[8] However, there are sharper formulations in which   is considered (but other values may be excluded, for example  ).[9]

Relevant corollaries of the Gagliardo-Nirenberg inequality edit

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   (so that the choice of   becomes irrelevant, and the same goes for the associated requirement  ) and the remaining parameters in such a way that
     
    and the other hypotheses are satisfied. The result reads then
     
    for any   such that  . In particular, setting   and   yields that  , namely the Sobolev conjugate exponent of  , and we have the embedding
     
    Notice that, in the embedding above, we also implicitly assume that   and hence the first exceptional case does not apply.
  • The Ladyzhenskaya inequality is a special case of the Gagliardo-Nirenberg inequality. Considering the most common cases, namely   and  , we have the former corresponding to the parameter choice
     
    yielding
     
    for any   The constant   is universal and can be proven to be  .[14] In three space dimensions, a slightly different choice of parameters is needed, namely
     
    yielding
     
    for any  . Here, it holds  .[14]
  • The Nash inequality, which was published by John Nash in 1958, is yet another result generalized by the Gagliardo-Nirenberg inequality. Indeed, choosing
     
    one gets
     
    which is oftentimes recast as
     
    or its squared version.[15][16]

Proof of the Gagliardo-Nirenberg inequality edit

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.[6]

The most delicate point concerns the limiting case  . In order to avoid the two exceptional cases, we further assume that   is finite and that  , so in particular  . The core of the proof is based on two proofs by induction.

Sketch of the proof of the Gagliardo-Nirenberg inequality[6]

Throughout the proof, given   and  , we shall assume that  . A double induction argument is applied to the couple of integers  , representing the orders of differentiation. The other parameters are constructed in such a way that they comply with the hypotheses of the theorem. As base case, we assume that the Gagliardo-Nirenberg inequality holds for   and   (hence  ). Here, in order for the inequality to hold, the remaining parameters should satisfy

 
The first induction step goes as follows. Assume the Gagliardo-Nirenberg inequality holds for some   strictly greater than   and   (hence  ). We are going to prove that it also holds for   and   (with  ). To this end, the remaining parameters   necessarily satisfy
 
Fix them as such. Then, let   be such that
 
From the base case, we can infer that
 
Now, from the two relations between the parameters, through some algebraic manipulations we arrive at
 
therefore the inequality with   applied to   implies
 
The two inequalities imply the sought Gagliardo-Nirenberg inequality, namely
 
The second induction step is similar, but allows   to change. Assume the Gagliardo-Nirenberg inequality holds for some pair   with   (hence  ). It is enough to prove that it also holds for   and   (with  ). Again, fix the parameters   in such a way that
 
and let   be such that
 
The inequality with   and   applied to   entails
 
Since, by the first induction step, we can assume the Gagliardo-Nirenberg inequality holds with   and  , we get
 
The proof is completed by combining the two inequalities. 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]

The Gagliardo-Nirenberg inequality in bounded domains edit

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  , but rather some given bounded, open and connected set   In the following, we also assume that   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. Precisely,

Theorem[17] (Gagliardo-Nirenberg in bounded domains) — Let   be a measurable, bounded, open and connected domain satisfying the cone condition. Let   be a positive extended real quantity. Let   and   be non-negative integers such that  . Furthermore, let   be a positive extended real quantity,   be real and   such that the relations

 
hold. Then,
 
where   such that   and   is arbitrary, with one exceptional case:
  1. if   and   is a non-negative integer, then the additional assumption   (notice the strict inequality) is needed.

In any case, the constant   depends on the parameters  , on the domain  , but not on  .

The necessity of a different formulation with respect to the case   is rather straightforward to prove. Indeed, since   has finite Lebesgue measure, any affine function belongs to   for every   (including  ). Of course, it holds much more: affine functions belong to   and all their derivatives of order greater than or equal to two are identically equal to zero in  . It can be easily seen that the Gagliardo-Nirenberg inequality for the case   fails to be true for any non constant affine function, since a contradiction is immediately achieved when   and   , 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  , then one can choose   and get

 
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  -norm. 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   for any finite  

Generalization to non-integer orders edit

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.[10][11] Furthermore, their results do not depend on the dimension   and allow real values of   and  , rather than integer. Here,   is either the full space, a half-space or a bounded and Lipschitz domain. If   and   is an extended real quantity, the space   is defined as follows

 
and if   we set
 
where   and   denote the integer part and the fractional part of  , respectively, i.e.  .[19] In this definition, there is the understanding that  , so that the usual Sobolev spaces are recovered whenever   is a positive integer. These spaces are often referred to as fractional Sobolev spaces. A generalization of the Gagliardo-Nirenberg inequality to these spaces reads

Theorem[20] (Brezis-Mironescu) — Let   be either the whole space, a half-space or a bounded Lipschitz domain. Let   be three positive extended real quantities and let   be non-negative real numbers. Furthermore, let   and assume that

 
hold. Then,
 
for any   if and only if
 
The constant   depends on the parameters  , on the domain  , but not on  .

For example, the parameter choice

 
gives the estimate
 
The validity of the estimate is granted, for instance, from the fact that  .

See also edit

References edit

  1. ^ Gagliardo, Emilio (August 14–21, 1958). Propriétés de certaines classes de fonctions de   variables (PDF). International Congress of Mathematicians (in French). Edinburgh. p. xxiv.
  2. ^ Nirenberg, Louis (August 14–21, 1958). Inequalities for derivatives (PDF). International Congress of Mathematicians. Edinburgh. p. xxvii.
  3. ^ Gagliardo, Emilio (1958). "Proprietà di alcune classi di funzioni in più variabili". Ricerche di Matematica (in Italian). 7 (1): 102–137.
  4. ^ Gagliardo, Emilio (1959). "Ulteriori proprietà di alcune classi di funzioni di più variabili". Ricerche di Matematica (in Italian). 8: 24–51.
  5. ^ a b c Nirenberg, Louis (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. 3 (13): 115–162.
  6. ^ a b c d Fiorenza, Alberto; Formica, Maria Rosaria; Roskovec, Tomáš; Soudský, Filip (2021). "Detailed Proof of Classical Gagliardo–Nirenberg Interpolation Inequality with Historical Remarks". Zeitschrift für Analysis und ihre Anwendungen. 40 (2): 217–236. arXiv:1812.04281. doi:10.4171/ZAA/1681. ISSN 0232-2064. S2CID 119708752.
  7. ^ Miranda, Carlo (1963). "Su alcune disuguaglianze integrali". Atti dell'Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali (in Italian). 8 (7): 1–14.
  8. ^ a b Nirenberg, Louis (1966). "On an extended interpolation inequality". Annali della Scuola Normale Superiore di Pisa. 3 (20): 733–737.
  9. ^ a b Soudský, Filip; Molchanova, Anastasia; Roskovec, Tomáš (2018). "Interpolation between Hölder and Lebesgue spaces with applications". Journal of Mathematical Analysis and Applications. 466 (1): 160–168. arXiv:1801.06865. doi:10.1016/j.jmaa.2018.05.067. S2CID 119577652.
  10. ^ a b Brezis, Haïm; Mironescu, Petru (2018). "Gagliardo–Nirenberg inequalities and non-inequalities: The full story". Annales de l'Institut Henri Poincaré C. 35 (5): 1355–1376. Bibcode:2018AIHPC..35.1355B. doi:10.1016/j.anihpc.2017.11.007. ISSN 0294-1449. S2CID 58891735.
  11. ^ a b Brezis, Haïm; Mironescu, Petru (2019-10-15). "Where Sobolev interacts with Gagliardo–Nirenberg". Journal of Functional Analysis. 277 (8): 2839–2864. doi:10.1016/j.jfa.2019.02.019. ISSN 0022-1236. S2CID 128179938.
  12. ^ Brezis, Haim (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer. doi:10.1007/978-0-387-70914-7. ISBN 978-0-387-70913-0.
  13. ^ Nirenberg, Louis (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. 3 (13): 125.
  14. ^ a b Galdi, Giovanni Paolo (2011). An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems. Springer Monographs in Mathematics (2nd ed.). Springer. p. 55. doi:10.1007/978-0-387-09620-9. ISBN 978-0-387-09619-3.
  15. ^ Nash, John (1958). "Continuity of solutions of parabolic and elliptic equations". American Journal of Mathematics. 80 (4): 931–954. Bibcode:1958AmJM...80..931N. doi:10.2307/2372841. JSTOR 2372841.
  16. ^ Bouin, Emeric; Dolbeault, Jean; Schmeiser, Christian (2020). "A variational proof of Nash's inequality" (PDF). Atti dell'Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. 31 (1): 211–223. doi:10.4171/RLM/886. S2CID 119668382.
  17. ^ Nirenberg, Louis (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. 3 (13): 126.
  18. ^ Brezis, Haim (2011). Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer. p. 233. doi:10.1007/978-0-387-70914-7. ISBN 978-0-387-70913-0.
  19. ^ Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico (2012). "Hitchhikerʼs guide to the fractional Sobolev spaces". Bulletin des Sciences Mathématiques. 136 (5): 524. arXiv:1104.4345. doi:10.1016/j.bulsci.2011.12.004. ISSN 0007-4497. S2CID 55443959.
  20. ^ Brezis, Haïm; Mironescu, Petru (2018). "Gagliardo–Nirenberg inequalities and non-inequalities: The full story". Annales de l'Institut Henri Poincaré C. 35 (5): 1356. Bibcode:2018AIHPC..35.1355B. doi:10.1016/j.anihpc.2017.11.007. ISSN 0294-1449. S2CID 58891735.