Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.Caffarelli, Luis A.; Li, YanYan. Preface [Dedicated to Louis Nirenberg on the occasion of his 85th birthday. Part I]. Discrete Contin. Dyn. Syst. 28 (2010), no. 2, i–ii. doi:10.3934/dcds.2010.28.2i
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.Yau, Shing-Tung. Perspectives on geometric analysis. Surveys in differential geometry. Vol. X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.
Biography Nirenberg was born in Hamilton, Ontario to Jewish immigrants from the Russian Empire. 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.Morto il grande matematico Louis Nirenberg
Nirenberg's work was widely recognized, including the following awards and honors: Bôcher Memorial Prize (1959) Elected member of the American Academy of Arts and Sciences (1965) Elected member of the United States National Academy of Sciences (1969) Crafoord Prize (1982) Jeffery–Williams Prize (1987) Elected member of the American Philosophical Society (1987) Steele Prize for Lifetime Achievement (1994)1994 Steele Prizes. Notices Amer. Math. Soc. 41 (1994), no. 8, 905–912. National Medal of Science (1995)Louis Nirenberg receives National Medal of Science. With contributions by Luis Caffarelli and Joseph J. Kohn. Notices Amer. Math. Soc. 43 (1996), no. 10, 1111–1116. Chern Medal (2010)2010 Chern Medal awarded. Notices Amer. Math. Soc. 57 (2010), no. 11, 1472–1474. Steele Prize for Seminal Contribution to Research (2014), with Luis Caffarelli and Robert Kohn, for their article on the Navier–Stokes equations Abel Prize (2015)
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.Leray, Jean. Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934), no. 1, 193–248. His work was later put into the setting of a boundary value problem by Eberhard Hopf.Hopf, Eberhard. Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4 (1951), 213–231.
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.Scheffer, Vladimir. Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math. 66 (1976), no. 2, 535–552. 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.Scheffer, Vladimir. Hausdorff measure and the Navier-Stokes equations. Comm. Math. Phys. 55 (1977), no. 2, 97–112. 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.Struwe, Michael. On partial regularity results for the Navier-Stokes equations. Comm. Pure Appl. Math. 41 (1988), no. 4, 437–458.Lin, Fanghua. A new proof of the Caffarelli-Kohn-Nirenberg theorem. Comm. Pure Appl. Math. 51 (1998), no. 3, 241–257. 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.Morrey, Charles B., Jr. On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43 (1938), no. 1, 126–166. 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.See the second page of . 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.Cheng, Shiu Yuen; Yau, Shing Tung. On the regularity of the Monge-Ampère equation . Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. Their work was based upon the relation via the Legendre transform to the Minkowski problem, which they had previously resolved by differential-geometric estimates.Cheng, Shiu Yuen; Yau, Shing Tung. On the regularity of the solution of the n-dimensional Minkowski problem. Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. 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,Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. which they were able to establish as sufficient by using Calabi's third-derivative estimates in the interior.Calabi, Eugenio. Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J. 5 (1958), 105–126. With Joseph Kohn, they found analogous results in the setting of the complex Monge−Ampère equation. 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. 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. 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.Morrey, Charles B., Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130 Springer-Verlag New York, Inc., New York 1966 ix+506 pp.