Canadian-American mathematician (1925–2020)
Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.
Nearly all of his work was in the field of partial differential equations. 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. 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.
Biography.
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.
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.
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. 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 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.[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 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.[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 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.
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.[CNS84] 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.[C+85] In such general situations, the Evans−Krylov theory 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.[CNS85] As a particular case of their new class of equations, they were able to partially resolve the boundary-value problem for 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.
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 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.[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.
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.[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 Henri Berestycki, Nirenberg used the Alexandrov−Bakelman−Pucci estimate to improve and modify the methods of Gidas−Ni−Nirenberg, significantly reducing the need to assume regularity of the domain.[BN91a] In an important result with Srinivasa Varadhan, Berestycki and Nirenberg continued the study of domains with no assumed regularity. For linear operators, they related the validity of the maximum principle to positivity of a first eigenvalue and existence of a first eigenfunction.[BNV94] With Luis Caffarelli, Berestycki and Nirenberg applied their results to symmetry of functions on cylindrical domains.[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. thesis.[BCN97b] They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of Maria Esteban and Pierre-Louis Lions.[BCN97a]
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.[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.
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.[JN61] 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. Among other applications, it is particularly fundamental to Jürgen Moser's Harnack inequality and subsequent work.
The John−Nirenberg inequality and the more general foundations