Product Code Database
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. Discrete Contin. Dyn. Syst. 28 (2010), no. 2, i–ii.
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.
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:
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.
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, 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.
The Monge-Ampère equation, in the form of prescribing the determinant of the Hessian matrix 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.
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.
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 manifold 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 to improve and modify the methods of Gidas−Ni−Nirenberg, significantly reducing the need to assume regularity of the domain. 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. With Luis Caffarelli, Berestycki and Nirenberg applied their results to symmetry of functions on cylindrical domains. 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.Savin, Ovidiu. Regularity of flat level sets in phase transitions. Ann. of Math. (2) 169 (2009), no. 1, 41–78.del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng. On De Giorgi's conjecture in dimension N≥9. Ann. of Math. (2) 174 (2011), no. 3, 1485–1569. They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of Maria Esteban and Pierre-Louis Lions.
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,Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der mathematischen Wissenschaften, 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. complex analysis,Garnett, John B. Bounded analytic functions. Revised first edition. Graduate Texts in Mathematics, 236. Springer, New York, 2007. xiv+459 pp. harmonic analysis,García-Cuerva, José; Rubio de Francia, José L. Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matemática, 104. North-Holland Publishing Co., Amsterdam, 1985. x+604 pp. Fourier analysis,Grafakos, Loukas. Modern Fourier analysis. Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. xvi+624 pp. and partial differential equations. Among other applications, it is particularly fundamental to Jürgen Moser's Harnack inequality and subsequent work.Moser, Jürgen On Harnack's theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577–591.Moser, Jürgen. A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 (1964), 101–134.
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. Among other results, they were able to establish that smooth maps which are close in BMO norm have the same topological degree, and hence that degree can be meaningfully defined for mappings of vanishing mean oscillation (VMO).
By adapting the Dirichlet energy, it is standard to recognize solutions of certain 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.Rabinowitz, Paul H. Free vibrations for a semilinear wave equation. Comm. Pure Appl. Math. 31 (1978), no. 1, 31–68. Part of their work involved small extensions of the standard mountain pass theorem and Palais-Smale condition, which have become standard in textbooks.Mawhin, Jean; Willem, Michel. Critical point theory and Hamiltonian systems. Applied Mathematical Sciences, 74. Springer-Verlag, New York, 1989. xiv+277 pp.Struwe, Michael. Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, 34. Springer-Verlag, Berlin, 2008. xx+302 pp.Willem, Michel. Minimax theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. x+162 pp. 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.
A fundamental contribution of Brezis and Nirenberg to critical point theory dealt with local minimizers. In principle, the choice of function space is highly relevant, and a function could minimize among smooth functions without minimizing among the broader class of Sobolev space. Making use of an earlier regularity result of Brezis and Tosio Kato, Brezis and Nirenberg ruled out such phenomena for a certain class of Dirichlet energy.Brézis, Haïm; Kato, Tosio. Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures Appl. (9) 58 (1979), no. 2, 137–151. Their work was later extended by Jesús García Azorero, Juan Manfredi, and Ireneo Peral.García Azorero, J. P.; Peral Alonso, I.; Manfredi, Juan J. Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 2 (2000), no. 3, 385–404.
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.
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.Nash, John. The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63 (1956), 20–63. 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 . His work was later simplified by Takaaki Nishida and used in an analysis of the Boltzmann equation.Nishida, Takaaki. A note on a theorem of Nirenberg. J. Differential Geometry 12 (1977), no. 4, 629–633Nishida, Takaaki. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm. Math. Phys. 61 (1978), no. 2, 119–148.
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 which are intrinsically flat. This can also be viewed as resolving a question on the isometric embedding of as hypersurfaces. Such questions and natural generalizations were later taken up by Cheng, Yau, and Harold Rosenberg, among others.Cheng, Shiu Yuen; Yau, Shing Tung. Hypersurfaces with constant scalar curvature. Math. Ann. 225 (1977), no. 3, 195–204.Rosenberg, Harold. Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117 (1993), no. 2, 211–239.
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 of Euclidean space. Geometric calculations show that solutions of certain semilinear Yamabe problem 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.
Articles.
|
|
