Louis Nirenberg

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 Henri Berestycki, Haïm Brezis, Luis Caffarelli, and 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.

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.

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.

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. In 1977, Shiu-Yuen Cheng and Shing-Tung Yau resolved the existence and interior regularity for the Monge-Ampère equation, showing in particular that if the determinant of the hessian of a function is smooth, then the function itself must be smooth as well. 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.

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.

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.

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. In collaboration with Shmuel Agmon and Douglis, Nirenberg proved boundary regularity for elliptic equations of arbitrary order. They later extended their results to elliptic systems of arbitrary order. With Morrey, Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic, extending to the boundary earlier known work. 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.

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 was born in Hamilton, Ontario to Ukrainian Jewish immigrants. He attended Baron Byng High School and McGill University, completing his BS in both mathematics and physics in 1945. Through a summer job at the 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.

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.

Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.

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.

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.