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. 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
Minkowski problem in two-dimensions, the
Gagliardo–Nirenberg interpolation inequality, the
Newlander-Nirenberg theorem in
complex geometry, and the development of pseudo-differential operators with
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. His work was later put into the setting of a
boundary value problem by
Eberhard Hopf. 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. 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. Such results are referred to as "partial regularity." Soon afterwards,
Luis Caffarelli,
Robert Kohn, and Nirenberg localized and sharpened Scheffer's analysis. 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
Michael Struwe, and a simplified version of Caffarelli−Kohn−Nirenberg's analysis was later found by
Fang-Hua Lin. In 2014, the
American Mathematical Society recognized Caffarelli−Kohn−Nirenberg's paper with the
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. Nirenberg, as part of his Ph.D. thesis, extended Morrey's results to the setting of fully nonlinear elliptic equations. 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
Monge-Ampère equation, in the form of prescribing the determinant of the
Hessian of a function, is one of the standard examples of a fully nonlinear elliptic equation. In an
invited lecture at the 1974
International Congress of Mathematicians, Nirenberg announced results obtained with
Eugenio Calabi on the
boundary-value problem for the Monge−Ampère equation, based upon boundary regularity estimates and a
method of continuity. However, they soon realized their proofs to be incomplete. Their work was based upon the relation via the
Legendre transform to the
Minkowski problem, which they had previously resolved by differential-geometric estimates. In particular, their work did not make use of boundary regularity, and their results left such questions unresolved. In collaboration with
Luis Caffarelli and
Joel Spruck, Nirenberg resolved such questions, directly establishing boundary regularity and using it to build a direct approach to the Monge−Ampère equation based upon the method of continuity. Calabi and Nirenberg had successfully demonstrated uniform control of the first two derivatives; the key for the method of continuity is the more powerful uniform
Hölder continuity of the second derivatives. Caffarelli, Nirenberg, and Spruck established a delicate version of this along the boundary, which they were able to establish as sufficient by using Calabi's third-derivative estimates in the interior. With
Joseph Kohn, they found analogous results in the setting of the complex Monge−Ampère equation. In such general situations, the Evans−Krylov theory 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.
Maximum principle and its applications In the case of
harmonic functions, the
maximum principle was known in the 1800s, and was used by
Carl Friedrich Gauss. In the early 1900s, complicated extensions to general second-order
elliptic partial differential equations were found by
Sergei Bernstein,
Leon Lichtenstein, and
Émile Picard; it was not until the 1920s that the simple modern proof was found by
Eberhard Hopf. In one of his earliest works, Nirenberg adapted Hopf's proof to second-order
parabolic partial differential equations, thereby establishing the
strong maximum principle in that context. 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. In the 1950s,
A.D. Alexandrov introduced an elegant "moving plane" reflection method, which he used as the context for applying the maximum principle to characterize the standard sphere as the only
closed hypersurface of
Euclidean space with
constant mean curvature. In 1971,
James Serrin utilized Alexandrov's technique to prove that highly symmetric solutions of certain second-order elliptic partial differential equations must be supported on symmetric domains. Nirenberg realized that Serrin's work could be reformulated so as to prove that solutions of second-order elliptic partial differential equations inherit symmetries of their domain and of the equation itself. Such results do not hold automatically, and it is nontrivial to identify which special features of a given problem are relevant. For example, there are many
harmonic functions on
Euclidean space which fail to be rotationally symmetric, despite the rotational symmetry of the
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. 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. 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
Henri Berestycki, Nirenberg used the Alexandrov−Bakelman−Pucci estimate They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of
Maria Esteban and
Pierre-Louis Lions.
Functional inequalities Nirenberg and
Emilio Gagliardo independently proved fundamental inequalities for
Sobolev spaces, now known as the
Gagliardo–Nirenberg–Sobolev inequality and the
Gagliardo–Nirenberg interpolation inequalities. 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. With
Luis Caffarelli and
Robert Kohn, Nirenberg would establish corresponding inequalities for certain weighted norms. Caffarelli, Kohn, and Nirenberg's norms were later investigated more fully in notable work by Florin Catrina and Zhi-Qiang Wang. Immediately following
Fritz John's introduction of the
bounded mean oscillation (BMO) function space in the theory of
elasticity, he and Nirenberg gave a further study of the space, proving in particular the "John−Nirenberg inequality," which constrains the size of the set on which a BMO function is far from its average value. Their work, which is an application of the
Calderon−Zygmund decomposition, has become a part of the standard mathematical literature. Expositions are contained in standard textbooks on probability, complex analysis, harmonic analysis, Fourier analysis, and partial differential equations. With
Haïm Brezis and
Guido Stampacchia, Nirenberg derived results extending both Fan's theory and Stampacchia's generalization of the
Lax-Milgram theorem. Their work has applications to the subject of
variational inequalities. By adapting the
Dirichlet energy, it is standard to recognize solutions of certain
wave equations as
critical points of functionals. With Brezis and
Jean-Michel Coron, Nirenberg found a novel functional whose critical points can be directly used to construct solutions of wave equations. They were able to apply the
mountain pass theorem to their new functional, thereby establishing the existence of periodic solutions of certain wave equations, extending a result of
Paul Rabinowitz. Part of their work involved small extensions of the standard mountain pass theorem and
Palais-Smale condition, which have become standard in textbooks. 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. Their work was later extended by Jesús García Azorero,
Juan Manfredi, and Ireneo Peral. 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. With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results.
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 :\frac{du}{dt}+Au=0 to the spectral properties of the operator
A. Applications include the study of rather general parabolic and elliptic-parabolic problems. 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. In
John Nash's work on the
isometric embedding problem, the key step is a small perturbation result, highly reminiscent of an
implicit function theorem; his proof used a novel combination of
Newton's method (in an infinitesimal form) with smoothing operators. Nirenberg was one of many mathematicians to put Nash's ideas into systematic and abstract frameworks, referred to as
Nash-Moser theorems. Nirenberg's formulation is particularly simple, isolating the basic analytic ideas underlying the analysis of most Nash-Moser iteration schemes. Within a similar framework, he proved an abstract form of the
Cauchy–Kowalevski theorem, as a particular case of a theorem on solvability of
ordinary differential equations in families of
Banach spaces. His work was later simplified by Takaaki Nishida and used in an analysis of the
Boltzmann equation.
Geometric problems Making use of his work on fully nonlinear elliptic equations, Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and
Minkowski problem in the field of
differential geometry. 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. Answering a question posed to Nirenberg by
Shiing-Shen Chern and
André Weil, Nirenberg and his doctoral student August Newlander proved what is now known as the
Newlander-Nirenberg theorem, which provides the precise algebraic condition under which an
almost complex structure arises from a holomorphic coordinate atlas. 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. The classical
Poincaré disk model assigns the metric of
hyperbolic space to the unit ball. Nirenberg and
Charles Loewner studied the more general means of naturally assigning a complete Riemannian metric to bounded
open subsets of
Euclidean space. 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. 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. 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==