Many of his contributions are now regarded as fundamental to the field, such as his strong maximum principle for second-order parabolic partial differential equations and the Newlander–Nirenberg theorem in complex geometry.

Following their discussion, Nirenberg was invited to enter graduate school at the Courant Institute of Mathematical Sciences at New York University.

He was the advisor of 45 PhD students, and published over 150 papers with a number of coauthors, including notable collaborations with Henri Berestycki, Haïm Brezis, Luis Caffarelli, and Yanyan Li, among many others.

With Basilis Gidas and Wei-Ming Ni he made innovative uses of the maximum principle to prove symmetry of many solutions of differential equations.

[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 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.

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.

[23][24] 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."

In the 1930s, Charles Morrey found the basic regularity theory of quasilinear elliptic partial differential equations for functions on two-dimensional domains.

[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.

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.

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.

[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.

[34] 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.

[35][36][37][38][39][40] 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.

[BCN96] They obtained in particular a partial resolution of a well-known conjecture of Ennio De Giorgi on translational symmetry, which was later fully resolved in Ovidiu Savin's Ph.D.

[BCN97b][41][42] They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of Maria Esteban and Pierre-Louis Lions.

[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.

With Brezis and Jean-Michel Coron, Nirenberg found a novel functional whose critical points can be directly used to construct solutions of wave equations.

[BCN80] 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.

[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.

[BN83] With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results.

[BCN94] 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.

[BN78a][BN78b] 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.

[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.

Nirenberg and Charles Loewner studied the more general means of naturally assigning a complete Riemannian metric to bounded open subsets of Euclidean space.

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.