Louis Nirenberg

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Louis Nirenberg
Steele Prize (1994, 2014)
National Medal of Science (1995)
Chern Medal (2010)
Abel Prize in Mathematics (2015)
Scientific career
FieldsMathematics
InstitutionsNew York University
Thesis The determination of a closed convex surface having given line elements  (1949)
Doctoral advisorJames Stoker
Doctoral students
Notes

Louis Nirenberg (February 28, 1925 – January 26, 2020) was a

Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.[2][3]

Nearly all of his work was in the field of

Newlander–Nirenberg theorem in complex geometry. He is regarded as a foundational figure in the field of geometric analysis, with many of his works being closely related to the study of complex analysis and differential geometry.[4]

Biography

Nirenberg was born in

National Research Council of Canada, he came to know Ernest Courant's wife Sara Paul. She spoke to Courant's father, the eminent mathematician Richard Courant, for advice on where Nirenberg should apply to study theoretical physics. Following their discussion, Nirenberg was invited to enter graduate school at the Courant Institute of Mathematical Sciences at New York University. In 1949, he obtained his doctorate in mathematics, under the direction of James Stoker. In his doctoral work, he solved the "Weyl problem" in differential geometry
, which had been a well-known open problem since 1916.

Following his doctorate, he became a professor at the Courant Institute, where he remained for the rest of his career. He was the advisor of 45 PhD students, and published over 150 papers with a number of coauthors, including notable collaborations with

Yanyan Li, among many others. He continued to carry out mathematical research until the age of 87. On January 26, 2020, Nirenberg died at the age of 94.[5][6][7]

Nirenberg's work was widely recognized, including the following awards and honors:

Mathematical achievements

Nirenberg is especially known for his collaboration with Shmuel Agmon and Avron Douglis in which they extended the Schauder theory, as previously understood for second-order elliptic partial differential equations, to the general setting of elliptic systems. With Basilis Gidas and Wei-Ming Ni he made innovative uses of the maximum principle to prove symmetry of many solutions of differential equations. The study of the BMO function space was initiated by Nirenberg and Fritz John in 1961; while it was originally introduced by John in the study of elastic materials, it has also been applied to games of chance known as martingales.[18] His 1982 work with Luis Caffarelli and Robert Kohn made a seminal contribution to the Navier–Stokes existence and smoothness, in the field of mathematical fluid mechanics.

Other achievements include the resolution of the

Joseph Kohn
.

Navier-Stokes equations

The

Navier-Stokes equations were developed in the early 1800s to model the physics of fluid mechanics. Jean Leray, in a seminal achievement in the 1930s, formulated an influential notion of weak solution for the equations and proved their existence.[19] His work was later put into the setting of a boundary value problem by Eberhard Hopf.[20]

A breakthrough came with work of Vladimir Scheffer in the 1970s. He showed that if a smooth solution of the Navier−Stokes equations approaches a singular time, then the solution can be extended continuously to the singular time away from, roughly speaking, a curve in space.[21] Without making such a conditional assumption on smoothness, he established the existence of Leray−Hopf solutions which are smooth away from a two-dimensional surface in spacetime.[22] Such results are referred to as "partial regularity." Soon afterwards, Luis Caffarelli, Robert Kohn, and Nirenberg localized and sharpened Scheffer's analysis.[CKN82] The key tool of Scheffer's analysis was an energy inequality providing localized integral control of solutions. It is not automatically satisfied by Leray−Hopf solutions, but Scheffer and Caffarelli−Kohn−Nirenberg established existence theorems for solutions satisfying such inequalities. With such "a priori" control as a starting point, Caffarelli−Kohn−Nirenberg were able to prove a purely local result on smoothness away from a curve in spacetime, improving Scheffer's partial regularity.

Similar results were later found by

Steele Prize for Seminal Contribution to Research, saying that their work was a "landmark" providing a "source of inspiration for a generation of mathematicians." The further analysis of the regularity theory of the Navier−Stokes equations is, as of 2021, a well-known open problem
.

Nonlinear elliptic partial differential equations

In the 1930s,

Charles Morrey found the basic regularity theory of quasilinear elliptic partial differential equations for functions on two-dimensional domains.[25] Nirenberg, as part of his Ph.D. thesis, extended Morrey's results to the setting of fully nonlinear elliptic equations.[N53a]
The works of Morrey and Nirenberg made extensive use of two-dimensionality, and the understanding of elliptic equations with higher-dimensional domains was an outstanding open problem.

The

Legendre transform to the Minkowski problem, which they had previously resolved by differential-geometric estimates.[28]
In particular, their work did not make use of boundary regularity, and their results left such questions unresolved.

In collaboration with

Hölder continuity of the second derivatives. Caffarelli, Nirenberg, and Spruck established a delicate version of this along the boundary,[29] which they were able to establish as sufficient by using Calabi's third-derivative estimates in the interior.[30] With Joseph Kohn, they found analogous results in the setting of the complex Monge−Ampère equation.[C+85] In such general situations, the Evans−Krylov theory[29]
is a more flexible tool than the computation-based calculations of Calabi.

Caffarelli, Nirenberg, and Spruck were able to extend their methods to more general classes of fully nonlinear elliptic partial differential equations, in which one studies functions for which certain relations between the hessian's eigenvalues are prescribed.

special Lagrangians
.

Linear elliptic systems

Nirenberg's most renowned work from the 1950s deals with "elliptic regularity." With Avron Douglis, Nirenberg extended the Schauder estimates, as discovered in the 1930s in the context of second-order elliptic equations, to general elliptic systems of arbitrary order.[DN55] In collaboration with Shmuel Agmon and Douglis, Nirenberg proved boundary regularity for elliptic equations of arbitrary order.[ADN59] They later extended their results to elliptic systems of arbitrary order.[ADN64] With Morrey, Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic, extending to the boundary earlier known work.[MN57] These contributions to elliptic regularity are now considered as part of a "standard package" of information, and are covered in many textbooks. The Douglis−Nirenberg and Agmon−Douglis−Nirenberg estimates, in particular, are among the most widely-used tools in elliptic partial differential equations.[31]

With

Yanyan Li, and motivated by composite materials in elasticity theory, Nirenberg studied linear elliptic systems in which the coefficients are Hölder continuous in the interior but possibly discontinuous on the boundary. Their result is that the gradient of the solution is Hölder continuous, with a L∞ estimate for the gradient which is independent of the distance from the boundary.[LN03]

Maximum principle and its applications

In the case of

strong maximum principle in that context.[N53b] As in the earlier work, such a result had various uniqueness and comparison theorems as corollaries. Nirenberg's work is now regarded as one of the foundations of the field of parabolic partial differential equations, and is ubiquitous across the standard textbooks.[35][36][37][38][39][40]

In the 1950s,

Laplacian
and of Euclidean space.

Nirenberg's first results on this problem were obtained in collaboration with Basilis Gidas and Wei-Ming Ni. They developed a precise form of Alexandrov and Serrin's technique, applicable even to fully nonlinear elliptic and parabolic equations.[GNN79] In a later work, they developed a version of the Hopf lemma applicable on unbounded domains, thereby improving their work in the case of equations on such domains.[GNN81] Their main applications deal with rotational symmetry. Due to such results, in many cases of geometric or physical interest, it is sufficient to study ordinary differential equations rather than partial differential equations.

Later, with

Maria Esteban and Pierre-Louis Lions.[BCN97a]

Functional inequalities

Nirenberg and

Gagliardo–Nirenberg interpolation inequalities.[N59] They are used ubiquitously throughout the literature on partial differential equations; as such, it has been of great interest to extend and adapt them to various situations. Nirenberg himself would later clarify the possible exponents which can appear in the interpolation inequality.[N66] With Luis Caffarelli and Robert Kohn, Nirenberg would establish corresponding inequalities for certain weighted norms.[CKN84] Caffarelli, Kohn, and Nirenberg's norms were later investigated more fully in notable work by Florin Catrina and Zhi-Qiang Wang.[43]

Immediately following

Harnack inequality and subsequent work.[48][49][29]

The John−Nirenberg inequality and the more general foundations of the BMO theory were worked out by Nirenberg and

Haïm Brézis in the context of maps between Riemannian manifolds.[BN95] Among other results, they were able to establish that smooth maps which are close in BMO norm have the same topological degree, and hence that degree can be meaningfully defined for mappings of vanishing mean oscillation (VMO)
.

Calculus of variations

In the setting of

Lax-Milgram theorem.[BNS72][52] Their work has applications to the subject of variational inequalities.[53]

By adapting the

Palais-Smale condition, which have become standard in textbooks.[55][56][57] In 1991, Brezis and Nirenberg showed how Ekeland's variational principle could be applied to extend the mountain pass theorem, with the effect that almost-critical points can be found without requiring the Palais−Smale condition.[BN91b][57]

A fundamental contribution of Brezis and Nirenberg to critical point theory dealt with local minimizers.[BN93] In principle, the choice of function space is highly relevant, and a function could minimize among smooth functions without minimizing among the broader class of Sobolev functions. Making use of an earlier regularity result of Brezis and Tosio Kato, Brezis and Nirenberg ruled out such phenomena for a certain class of Dirichlet-type functionals.[58] Their work was later extended by Jesús García Azorero, Juan Manfredi, and Ireneo Peral.[59]

In one of Nirenberg's most widely cited papers, he and Brézis studied the Dirichlet problem for Yamabe-type equations on Euclidean spaces, following part of Thierry Aubin's work on the Yamabe problem.[BN83] With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results.[BCN94]

Nonlinear functional analysis

Agmon and Nirenberg made an extensive study of ordinary differential equations in Banach spaces, relating asymptotic representations and the behavior at infinity of solutions to

to the spectral properties of the operator A. Applications include the study of rather general parabolic and elliptic-parabolic problems.[AN63]

Brezis and Nirenberg gave a study of the perturbation theory of nonlinear perturbations of noninvertible transformations between Hilbert spaces; applications include existence results for periodic solutions of some semilinear wave equations.[BN78a][BN78b]

In

Cauchy–Kowalevski theorem, as a particular case of a theorem on solvability of ordinary differential equations in families of Banach spaces.[N72] His work was later simplified by Takaaki Nishida and used in an analysis of the Boltzmann equation.[61][62]

Geometric problems

Making use of his work on fully nonlinear elliptic equations[N53a], Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem in the field of differential geometry.[N53c] The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two-dimensional sphere into three-dimensional Euclidean space, while the latter asks for closed surfaces in three-dimensional Euclidean space for which the Gauss map prescribes the Gaussian curvature. The key is that the "Darboux equation" from surface theory is of Monge−Ampère type, so that Nirenberg's regularity theory becomes useful in the method of continuity. John Nash's well-known isometric embedding theorems, established soon afterwards, have no apparent relation to the Weyl problem, which deals simultaneously with high-regularity embeddings and low codimension.[63][60] Nirenberg's work on the Minkowski problem was extended to Riemannian settings by Aleksei Pogorelov. In higher dimensions, the Minkowski problem was resolved by Shiu-Yuen Cheng and Shing-Tung Yau.[28] Other approaches to the Minkowski problem have developed from Caffarelli, Nirenberg, and Spruck's fundamental contributions to the theory of nonlinear elliptic equations.[CNS85]

In one of his very few articles not centered on analysis, Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat.[HN59] This can also be viewed as resolving a question on the isometric embedding of flat manifolds as hypersurfaces. Such questions and natural generalizations were later taken up by Cheng, Yau, and Harold Rosenberg, among others.[64][65]

Answering a question posed to Nirenberg by

Newlander-Nirenberg theorem, which provides the precise algebraic condition under which an almost complex structure arises from a holomorphic coordinate atlas.[NN57] The Newlander-Nirenberg theorem is now considered as a foundational result in complex geometry, although the result itself is far better known than the proof, which is not usually covered in introductory texts, as it relies on advanced methods in partial differential equations. Nirenberg and Joseph Kohn, following earlier work by Kohn, studied the ∂-Neumann problem on pseudoconvex domains, and demonstrated the relation of the regularity theory to the existence of subelliptic estimates for the ∂ operator.[KN65b]

The classical

open subsets of Euclidean space.[LN74] Geometric calculations show that solutions of certain semilinear Yamabe-type equations
can be used to define metrics of constant scalar curvature, and that the metric is complete if the solution diverges to infinity near the boundary. Loewner and Nirenberg established existence of such solutions on certain domains. Similarly, they studied a certain Monge−Ampère equation with the property that, for any negative solution extending continuously to zero at the boundary, one can define a complete Riemannian metric via the hessian. These metrics have the special property of projective invariance, so that projective transformation from one given domain to another becomes an isometry of the corresponding metrics.

Pseudo-differential operators

Joseph Kohn and Nirenberg introduced the notion of pseudo-differential operators.[KN65a] Nirenberg and François Trèves investigated the famous Lewy's example for a non-solvable linear PDE of second order, and discovered the conditions under which it is solvable, in the context of both partial differential operators and pseudo-differential operators.[NT63][NT70] Their introduction of local solvability conditions with analytic coefficients has become a focus for researchers such as R. Beals, C. Fefferman, R.D. Moyer, Lars Hörmander, and Nils Dencker
who solved the pseudo-differential condition for Lewy's equation. This opened up further doors into the local solvability of linear partial differential equations.

Major publications

Books and surveys.

N73.
Nirenberg, Louis (1973). Lectures on linear partial differential equations. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics. Vol. 17. Providence, RI:
Zbl 0267.35001
.
N94.
Nirenberg, Louis (1994). "Partial differential equations in the first half of the century". In
Zbl 0807.01017
.
N01.
Nirenberg, Louis (2001). Topics in nonlinear functional analysis. Courant Lecture Notes in Mathematics. Vol. 6. Chapter 6 by
Zbl 0286.47037
.
N18.
Nirenberg, Louis (2018). Lectures on differential equations and differential geometry. Classical Topics in Mathematics. Vol. 7. With a preface by
Zbl 1465.35001
.

Articles.

N53a.
Nirenberg, Louis (1953). "On nonlinear elliptic partial differential equations and Hölder continuity".
doi:10.1002/cpa.3160060304
)
N53b.
Nirenberg, Louis (1953). "A strong maximum principle for parabolic equations".
Zbl 0050.09601
.
N53c.
Nirenberg, Louis (1953). "The Weyl and Minkowski problems in differential geometry in the large".
Zbl 0051.12402
.
DN55.
Douglis, Avron; Nirenberg, Louis (1955). "Interior estimates for elliptic systems of partial differential equations".
Zbl 0066.08002
.
N55.
Nirenberg, Louis (1955). "Remarks on strongly elliptic partial differential equations".
Zbl 0067.07602
.
MN57.
NN57.
Newlander, A.; Nirenberg, L. (1957). "Complex analytic coordinates in almost complex manifolds".
Zbl 0079.16102
.
ADN59.
HN59.
N59.
Nirenberg, L. (1959). "On elliptic partial differential equations". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III. 13 (2): 115–162.
Zbl 0088.07601
.
JN61.
AN63.
NT63.
Nirenberg, L.;
Zbl 0117.06104
.
ADN64.
KN65a.
KN65b.
N66.
Nirenberg, L. (1966). "An extended interpolation inequality". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie III. 20 (4): 733–737.
Zbl 0163.29905
.
KN67.
NT70.
Nirenberg, Louis;
Zbl 0191.39103
.
BNS72.
NW73.
LN74.
KN77.
BN78a.
BN78b.
GNN79.
BCN80.
GNN81.
CKN82.
BN83.
CKN84.
C+85.
CNS85.
BN91a.
BN91b.
BN92.
BN93.
BCN94.
BNV94.
BCN95.
BN95.
BCN96.
BCN97a.
BCN97b.
LN03.

References

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External links

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