# Louis Nirenberg

Louis Nirenberg
Louis Nirenberg in 1975
Born28 February 1925
Died26 January 2020 (aged 94)
Alma materMcGill University (BS, 1945)
New York University (PhD, 1950)
Known forPartial differential equations
Gagliardo–Nirenberg interpolation inequality
Gagliardo–Nirenberg–Sobolev inequality
Bounded mean oscillation (John–Nirenberg space)
AwardsBôcher Memorial Prize (1959)
Crafoord Prize (1982)
Steele Prize (1994, 2014)
National Medal of Science (1995)
Chern Medal (2010)
Abel Prize in Mathematics (2015)
Scientific career
FieldsMathematics
InstitutionsNew York University
ThesisThe determination of a closed convex surface having given line elements (1949)
Doctoral students
Notes

Louis Nirenberg (28 February 1925 – 26 January 2020) was a Canadian-American mathematician, considered one of the most outstanding mathematicians of the 20th century.[1][2]

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 proof of the strong maximum principle for second-order parabolic partial differential equations. 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.[3]

He 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.[4] His 1982 work with Luis Caffarelli and Robert Kohn was described by Charles Fefferman in 2002 as "about the best that's been done" on the Millennium Prize problem of Navier–Stokes existence and smoothness, in the field of mathematical fluid mechanics.[1]

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.

## Biography

Nirenberg was born in Hamilton, Ontario to Ukrainian immigrants. He attended Baron Byng High School and McGill University, completing his B.S. 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 Ph.D. 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 26 January 2020, Nirenberg died at the age of 94.[5][6][7]

## Mathematical achievements

### 1950s

Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem of differential geometry. The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two-dimensional sphere into three-dimensional Euclidean space, while the latter asks for closed surfaces in three-dimensional Euclidean space of prescribed Gaussian curvature. The now-standard approach to these problems is through the theory of the Monge-Ampère equation, which is a fully nonlinear elliptic partial differential equation. Nirenberg made novel contributions to the theory of such equations in the setting of two-dimensional domains, building on the earlier 1938 work of Charles Morrey. Nirenberg's work on the Minkowski problem was significantly extended by Aleksei Pogorelov, Shiu-Yuen Cheng, and Shing-Tung Yau, among other authors. In a separate contribution to differential geometry, Nirenberg and Philip Hartman characterized the cylinders in Euclidean space as the only complete hypersurfaces which are intrinsically flat.

In the same year as his resolution of the Weyl and Minkowski problems, Nirenberg made a major contribution to the understanding of the maximum principle, proving the strong maximum principle for second-order parabolic partial differential equations. This is now regarded as one of the most fundamental results in this setting.[11]

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 Douglis and Shmuel Agmon, Nirenberg extended these estimates up to the boundary. 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.[12]

In 1957, 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 a precise 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.

In his 1959 survey on elliptic differential equations, Nirenberg proved (independently of Emilio Gagliardo) what is now known as the Gagliardo-Nirenberg interpolation inequalities for the Sobolev spaces. A later work by Nirenberg, in 1966, clarified the possible exponents which can appear in these inequalities. More recent work by other authors has extended the Gagliardo-Nirenberg inequalities to the fractional Sobolev spaces.

### 1960s

Immediately following Fritz John's introduction of the BMO function space in the theory of elasticity, John and Nirenberg gave a further study of the space, with a particular functional inequality, now known as the John-Nirenberg inequality, which has become basic in the field of harmonic analysis. It characterizes how quickly a BMO function deviates from its average; the proof is a classic application of the Calderon-Zygmund decomposition.

Nirenberg and François Trèves investigated the famous Lewy's example for a non-solvable linear PDE of second order, and discovered the conditions under which it is solvable, in the context of both partial differential operators and pseudo-differential operators. Their introduction of local solvability conditions with analytic coefficients has become a focus for researchers such as R. Beals, C. Fefferman, R.D. Moyer, Lars Hörmander, and Nils Dencker who solved the pseudo-differential condition for Lewy's equation. This opened up further doors into the local solvability of linear partial differential equations.

Nirenberg and J.J. 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.

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

${\displaystyle {\frac {du}{dt}}+Au=0}$

to the spectral properties of the operator A. Applications include the study of rather general parabolic and elliptic-parabolic problems.

### 1970s

In the 1960s, A.D. Aleksandrov introduced an elegant "sliding plane" reflection method, which he used to apply the maximum principle in proving that the only closed hypersurface of Euclidean space which has constant mean curvature is the round sphere. In collaboration with Basilis Gidas and Wei-Ming Ni, Nirenberg gave an extensive study of how this method applies to prove symmetry of solutions of certain symmetric second-order elliptic partial differential equations. A sample result is that if u is a positive function on a ball with zero boundary data and with Δu + f(u) = 0 on the interior of the ball, then u is rotationally symmetric. In a later 1981 paper, they extended this work to symmetric second-order elliptic partial differential equations on all of n. These two papers are among Nirenberg's most widely cited, due to the flexibility of their techniques and the corresponding generality of their results. Due to Gidas, Ni, and Nirenberg's results, in many cases of geometric or physical interest, it is sufficient to study ordinary differential equations rather than partial differential equations. The resulting problems were taken up in a number of influential works by Ni, Henri Berestycki, Pierre-Louis Lions, and others.

Nirenberg and Charles Loewner studied the means of naturally assigning a complete Riemannian metric to bounded open subsets of Euclidean space, modeled on the classical assignment of hyperbolic space to the unit ball, via the unit ball model. They showed that if Ω is a bounded open subset of 2 with smooth and strictly convex boundary, then the Monge-Ampère equation

${\displaystyle \operatorname {det} D^{2}u={\frac {1}{|u|^{4}}}}$

has a unique smooth negative solution which extends continuously to zero on the boundary Ω. The geometric significance of this result is that 1/uD2u then defines a complete Riemannian metic on Ω. In the special case that Ω is a ball, this recovers the hyperbolic metric. Loewner and Nirenberg also studied the method of conformal deformation, via the Yamabe equation

${\displaystyle \Delta u=cu^{(n+2)/(n-2)}}$

for a constant c. They showed that for certain Ω, this Yamabe equation has a unique solution which diverges to infinity at the boundary. The geometric significance of such a solution is that u2/(n − 2)gEuc is then a complete Riemannian metric on Ω which has constant scalar curvature.

In other work, Haïm Brezis, Guido Stampacchia, and Nirenberg gave an extension of Ky Fan's topological minimax principle to noncompact settings. 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.

### 1980s

Luis Caffarelli, Robert Kohn, and Nirenberg studied the three-dimensional incompressible Navier-Stokes equations, showing that the set of spacetime points at which weak solutions fail to be differentiable must, roughly speaking, fill less space than a curve. This is known as a "partial regularity" result. In his description of the conjectural regularity of the Navier-Stokes equations as a Millennium prize problem, Charles Fefferman refers to Caffarelli-Kohn-Nirenberg's result as the "best partial regularity theorem known so far" on the problem. As a by-product of their work on the Navier-Stokes equations, Caffarelli, Kohn, and Nirenberg (in a separate paper) extended Nirenberg's earlier work on the Gagliardo-Nirenberg interpolation inequality to certain weighted norms.

In 1977, Shiu-Yuen Cheng and Shing-Tung Yau had resolved the interior regularity for the Monge-Ampère equation, showing in particular that if the right-hand side is smooth, then the solution must be smooth as well. In 1984, Caffarelli, Joel Spruck, and Nirenberg used different methods to extend Cheng and Yau's results to the case of boundary regularity. They were able to extend their study to a more general class of fully nonlinear elliptic partial differential equations, in which solutions are determined by algebraic relations on the eigenvalues of the matrix of second derivatives. With J.J. Kohn, they also found analogous results in the setting of the complex Monge-Ampère equation.

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.

### 1990s

The moving plane method of Aleksandrov, as extended in 1979 by Gidas, Ni, and Nirenberg, is studied further in joint works by Berestycki, Caffarelli, and Nirenberg. The primary theme is to understand when a solution of Δu+f(u)=0, with Dirichlet data on a cylinder, necessarily inherits a cylindrical symmetry.

In 1991, Brezis and Nirenberg applied the Ekeland variational principle to extend the mountain pass lemma. In 1993, they made a fundamental contributions to critical point theory in showing (with some contextual assumptions) that a local minimizer of

${\displaystyle \int _{\Omega }{\Big (}|\nabla u|^{2}-\int _{0}^{u(x)}f(x,t)\,dt{\Big )}\,dx}$

in the C1 topology is also a local minimizer in the W1,2 topology. In 1995, they used density theorems to extend the notion of topological degree from continuous mappings to the class of VMO mappings.

With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results. These can be viewed as developments of Brezis and Nirenberg's fundamental paper from 1983.

In an important result with Berestycki and Srinivasa Varadhan, Nirenberg extended the classically-known results on the first eigenvalue of second-order elliptic operators to settings where the boundary of the domain is not differentiable.

In 1992, Berestycki and Nirenberg gave a complete study of the existence of traveling-wave solutions of reaction-diffusion equations in which the spatial domain is cylindrical, i.e. of the form ℝ×Ω'.

### 2000s

With Yanyan Li, and motivated by composite materials in elasticity theory, Nirenberg studied 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.

## Books and surveys

• Louis Nirenberg. Lectures on linear partial differential equations. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17. American Mathematical Society, Providence, R.I., 1973. v+58 pp.
• Louis Nirenberg. Topics in nonlinear functional analysis. Chapter 6 by E. Zehnder. Notes by R. A. Artino. Revised reprint of the 1974 original. Courant Lecture Notes in Mathematics, 6. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001. xii+145 pp. ISBN 0-8218-2819-3
• Louis Nirenberg. Lectures on differential equations and differential geometry. With a preface by Shiu-Yuen Cheng and Lizhen Ji. CTM. Classical Topics in Mathematics, 7. Higher Education Press, Beijing, 2018. ix+174 pp. ISBN 978-7-04-050302-9
• Nirenberg, L. On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 13 (1959), 115–162.
• Partial differential equations in the first half of the century, in Jean-Paul Pier Development of mathematics 1900–1950, Birkhäuser 1994

## Major publications

• Nirenberg, Louis. A strong maximum principle for parabolic equations. Comm. Pure Appl. Math. 6 (1953), 167–177.
• Nirenberg, Louis. The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6 (1953), 337–394.
• Douglis, Avron; Nirenberg, Louis. Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955), 503–538.
• Morrey, C.B., Jr.; Nirenberg, L. On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–290.
• Newlander, A.; Nirenberg, L. Complex analytic coordinates in almost complex manifolds. Ann. of Math. (2) 65 (1957), 391–404.
• Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959), 623–727.
• Hartman, Philip; Nirenberg, Louis. On spherical image maps whose Jacobians do not change sign. Amer. J. Math. 81 (1959), 901–920.
• John, F.; Nirenberg, L. On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14 (1961), 415–426.
• Agmon, S.; Nirenberg, L. Properties of solutions of ordinary differential equations in Banach space. Comm. Pure Appl. Math. 16 (1963), 121–239.
• Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17 (1964), 35–92.
• Kohn, J.J.; Nirenberg, L. Non-coercive boundary value problems. Comm. Pure Appl. Math. 18 (1965), 443–492.
• Nirenberg, L. An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 20 (1966), 733–737.
• Brézis, H.; Nirenberg, L.; Stampacchia, G. A remark on Ky Fan's minimax principle. Boll. Un. Mat. Ital. (4) 6 (1972), 293–300.
• Loewner, Charles; Nirenberg, Louis. Partial differential equations invariant under conformal or projective transformations. Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 245–272. Academic Press, New York, 1974.
• Brézis, H.; Nirenberg, L. Characterizations of the ranges of some nonlinear operators and applications to boundary value problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 225–326.
• Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979), no. 3, 209–243.
• Gidas, B.; Ni, Wei Ming; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in Rn. Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
• Caffarelli, L.; Kohn, R.; Nirenberg, L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35 (1982), no. 6, 771–831.
• Brézis, Haïm; Nirenberg, Louis. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.
• Caffarelli, L.; Kohn, R.; Nirenberg, L. First order interpolation inequalities with weights. Compositio Math. 53 (1984), no. 3, 259–275.
• Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation. Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402.
• Caffarelli, L.; Kohn, J.J.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge-Ampère, and uniformly elliptic, equations. Comm. Pure Appl. Math. 38 (1985), no. 2, 209–252.
• Caffarelli, L.; Nirenberg, L.; Spruck, J. The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), no. 3-4, 261–301.
• Berestycki, H.; Nirenberg, L. On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), no. 1, 1–37.
• Brezis, Haïm; Nirenberg, Louis. Remarks on finding critical points. Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939–963.
• Berestycki, Henri; Nirenberg, Louis. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), no. 5, 497–572.
• Brezis, Haïm; Nirenberg, Louis. H1 versus C1 local minimizers. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 5, 465–472.
• Berestycki, H.; Capuzzo-Dolcetta, I.; Nirenberg, L. Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78.
• Berestycki, H.; Nirenberg, L.; Varadhan, S.R.S. The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47 (1994), no. 1, 47–92.
• Berestycki, Henri; Capuzzo-Dolcetta, Italo; Nirenberg, Louis. Variational methods for indefinite superlinear homogeneous elliptic problems. NoDEA Nonlinear Differential Equations Appl. 2 (1995), no. 4, 553–572.
• Brezis, H.; Nirenberg, L. Degree theory and BMO. I. Compact manifolds without boundaries. Selecta Math. (N.S.) 1 (1995), no. 2, 197–263.
• Berestycki, H.; Caffarelli, L.A.; Nirenberg, L. Monotonicity for elliptic equations in unbounded Lipschitz domains. Comm. Pure Appl. Math. 50 (1997), no. 11, 1089–1111.
• Berestycki, Henri; Caffarelli, Luis; Nirenberg, Louis. Further qualitative properties for elliptic equations in unbounded domains. Dedicated to Ennio De Giorgi. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), no. 1-2, 69–94 (1998).
• Li, Yanyan; Nirenberg, Louis. Estimates for elliptic systems from composite material. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925.
• Li, Yanyan; Nirenberg, Louis. The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58 (2005), no. 1, 85–146.
• Li, Yanyan; Nirenberg, Louis. A geometric problem and the Hopf lemma. II. Chinese Ann. Math. Ser. B 27 (2006), no. 2, 193–218.
• Caffarelli, L.; Li, Yanyan, Nirenberg, Louis. Some remarks on singular solutions of nonlinear elliptic equations III: viscosity solutions including parabolic operators. Comm. Pure Appl. Math. 66 (2013), no. 1, 109–143.

## References

1. ^ a b Allyn Jackson (March 2002). "Interview with Louis Nirenberg" (PDF). Notices of the AMS. 49 (4): 441–449. Archived from the original (PDF) on 3 March 2016. Retrieved 26 March 2015.
2. ^ 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
3. ^ Yau, Shing-Tung. Perspectives on geometric analysis. Surveys in differential geometry. Vol. X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006.
4. ^ "John F. Nash Jr. and Louis Nirenberg share the Abel Prize". The Abel Prize. 25 March 2015. Retrieved 26 March 2015.
5. ^ Morto il grande matematico Louis Nirenberg (in Italian)
6. ^ Chang, Kenneth (31 January 2020). "Louis Nirenberg, 'One of the Great Mathematicians,' Dies at 94". New York Times. Retrieved 19 February 2020.
7. ^ Shields, Brit; Barany, Michael J. (17 February 2020). "Louis Nirenberg (1925–2020)". Nature. Retrieved 19 February 2020.
8. ^ 1994 Steele Prizes. Notices Amer. Math. Soc. 41 (1994), no. 8, 905–912.
9. ^ 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.
10. ^ 2010 Chern Medal awarded. Notices Amer. Math. Soc. 57 (2010), no. 11, 1472–1474.
11. ^ Evans, Lawrence C. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. xxii+749 pp. ISBN 978-0-8218-4974-3
12. ^ 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.