Louis Nirenberg (February 28, 1925 – January 26, 2020) was a
Canadian-American
Canadian Americans is a term that can be applied to American citizens whose ancestry is wholly or partly Canadian, or citizens of either country that hold dual citizenship.
The term ''Canadian'' can mean a nationality or an ethnicity. Canadia ...
mathematician, considered one of the most outstanding
mathematicians
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
One ...
of the 20th century.
Nearly all of his work was in the field of
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
. Many of his contributions are now regarded as fundamental to the field, such as his
strong maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
for second-order parabolic partial differential equations and the
Newlander-Nirenberg theorem
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...
in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
. He is regarded as a foundational figure in the field of
geometric analysis
Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations (PDEs), are used to establish new results in differential geometry and differential topology. The use of l ...
, with many of his works being closely related to the study of
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
.
Biography
Nirenberg was born in
Hamilton, Ontario
Hamilton is a port city in the Canadian province of Ontario. Hamilton has a population of 569,353, and its census metropolitan area, which includes Burlington and Grimsby, has a population of 785,184. The city is approximately southwest of T ...
to
Ukrainian Jewish
The history of the Jews in Ukraine dates back over a thousand years; Jewish communities have existed in the territory of Ukraine from the time of the Kievan Rus' (late 9th to mid-13th century). Some of the most important Jewish religious and ...
immigrants. He attended
Baron Byng High School
Baron Byng High School was an English-language public high school on Saint Urbain Street in Montreal, Quebec, opened by Governor General of Canada Julian Byng, 1st Viscount Byng of Vimy in 1921. The school was attended largely by working-class J ...
and
McGill University
McGill University (french: link=no, Université McGill) is an English-language public research university located in Montreal, Quebec, Canada. Founded in 1821 by royal charter granted by King George IV,Frost, Stanley Brice. ''McGill Universit ...
, completing his
BS in both mathematics and
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
in 1945. Through a summer job at the
National Research Council of Canada
The National Research Council Canada (NRC; french: Conseil national de recherches Canada) is the primary national agency of the Government of Canada dedicated to science and technology research & development. It is the largest federal research ...
, he came to know
Ernest Courant
Ernest Courant (March 26, 1920 – April 21, 2020) was an American accelerator physicist and a fundamental contributor to modern large-scale particle accelerator concepts. His most notable discovery was his 1952 work with Milton S. Livingston a ...
's wife Sara Paul. She spoke to Courant's father, the eminent mathematician
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
, 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
The Courant Institute of Mathematical Sciences (commonly known as Courant or CIMS) is the mathematics research school of New York University (NYU), and is among the most prestigious mathematics schools and mathematical sciences research cente ...
at
New York University
New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then-Secretary of the Treasury Albert Gallatin.
In 1832, the ...
. 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
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, 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
Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations.
Biography
Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, wh ...
,
Luis Caffarelli
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
, 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:
*
Bôcher Memorial Prize
The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). It is awarded every three years (formerly every five year ...
(1959)
* Elected member of the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...
(1965)
* Elected member of the United States
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...
(1969)
*
Crafoord Prize
The Crafoord Prize is an annual science prize established in 1980 by Holger Crafoord, a Swedish industrialist, and his wife Anna-Greta Crafoord. The Prize is awarded in partnership between the Royal Swedish Academy of Sciences and the Crafoord Foun ...
(1982)
*
Jeffery–Williams Prize The Jeffery–Williams Prize is a mathematics award presented annually by the Canadian Mathematical Society. The award is presented to individuals in recognition of outstanding contributions to mathematical research. The first award was presen ...
(1987)
* Elected member of the
American Philosophical Society
The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
(1987)
*
Steele Prize for Lifetime Achievement (1994)
*
National Medal of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral and social scienc ...
(1995)
*
Chern Medal
The Chern Medal is an international award recognizing outstanding lifelong achievement of the highest level in the field of mathematics. The prize is given at the International Congress of Mathematicians (ICM), which is held every four years.
...
(2010)
*
Steele Prize for Seminal Contribution to Research (2014), with
Luis Caffarelli
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
and
Robert Kohn, for their article on the Navier-Stokes equations
*
Abel Prize
The Abel Prize ( ; no, Abelprisen ) is awarded annually by the King of Norway to one or more outstanding mathematicians. It is named after the Norwegian mathematician Niels Henrik Abel (1802–1829) and directly modeled after the Nobel Prizes. ...
(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
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
to prove
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
of many solutions of differential equations. The study of the
BMO function space was initiated by Nirenberg and
Fritz John
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a ...
in 1961; while it was originally introduced by John in the study of
elastic materials, it has also been applied to
games of chance
A game of chance is in contrast with a game of skill. It is a game whose outcome is strongly influenced by some randomizing device. Common devices used include dice, spinning tops, playing cards, roulette wheels, or numbered balls drawn from ...
known as
martingales.
His 1982 work with
Luis Caffarelli
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
and
Robert Kohn made a seminal contribution to the
Navier–Stokes existence and smoothness, in the field of mathematical
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
.
Other achievements include the resolution of the
Minkowski problem
In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface ''S'' whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly posit ...
in two-dimensions, the
Gagliardo–Nirenberg interpolation inequality, the
Newlander-Nirenberg theorem
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...
in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, 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
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
.
Jean Leray
Jean Leray (; 7 November 1906 – 10 November 1998) was a French mathematician, who worked on both partial differential equations and algebraic topology.
Life and career
He was born in Chantenay-sur-Loire (today part of Nantes). He studied at Éc ...
, in a seminal achievement in the 1930s, formulated an influential notion of
weak solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisel ...
for the equations and proved their existence. His work was later put into the setting of a
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
by
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who ...
.
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
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
,
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
Michael Struwe (born 6 October 1955 in Wuppertal) is a German mathematician who specializes in calculus of variations and nonlinear partial differential equations. He won the 2012 Cantor medal from the Deutsche Mathematiker-Vereinigung for "o ...
, and a simplified version of Caffarelli−Kohn−Nirenberg's analysis was later found by
Fang-Hua Lin. In 2014, the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
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 equation
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
wher ...
s 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.
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
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...
, Nirenberg announced results obtained with
Eugenio Calabi
Eugenio Calabi (born 11 May 1923) is an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics, Emeritus, at the University of Pennsylvania, specializing in differential geometry, partial differential equations and ...
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
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
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
In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
to the
Minkowski problem
In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface ''S'' whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly posit ...
, 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
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
and
Joel Spruck
Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University ...
, 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 Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
* Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a modu ...
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. 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.
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.
Maximum principle and its applications
In the case of
harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s, the
maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
was known in the 1800s, and was used by
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
. In the early 1900s, complicated extensions to general second-order
elliptic partial differential equations
Second-order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic. Any second-order linear PDE in two variables can be written in the form
:Au_ + 2Bu_ + Cu_ + Du_x + Eu_y + Fu +G= 0,\,
whe ...
were found by
Sergei Bernstein
Sergei Natanovich Bernstein (russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as ; 5 March 1880 – 26 October 1968) was a Ukrainian and Russian mathematician of Jewish origin known for contributions to parti ...
,
Leon Lichtenstein
Leon Lichtenstein (16 May 1878 – 21 August 1933) was a Polish-German mathematician, who made contributions to the areas of differential equations, conformal mapping, and potential theory. He was also interested in theoretical physics, publish ...
, and
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at th ...
; it was not until the 1920s that the simple modern proof was found by
Eberhard Hopf
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a mathematician and astronomer, one of the founding fathers of ergodic theory and a pioneer of bifurcation theory who ...
. In one of his earliest works, Nirenberg adapted Hopf's proof to second-order
parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE). Parabolic PDEs are used to describe a wide variety of time-dependent phenomena, including heat conduction, particle diffusion, and pricing of derivati ...
s, thereby establishing the
strong maximum principle
In the mathematical fields of partial differential equations and geometric analysis, the maximum principle is any of a collection of results and techniques of fundamental importance in the study of elliptic and parabolic differential equations.
...
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.
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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
with
constant mean curvature
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature.Carl Johan Lejdfors, Surfaces of Constant Mean Curvature. Master’s thesis Lund University, Centre for Mathematical Sciences Mathematics 2 ...
. In 1971,
James Serrin
James Burton Serrin (1 November 1926, Chicago, Illinois – 23 August 2012, Minneapolis, Minnesota) was an American mathematician, and a professor at University of Minnesota.
Life
He received his doctorate from Indiana University in 1951 under t ...
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 function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is,
: \f ...
s on
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
which fail to be rotationally symmetric, despite the rotational symmetry of the
Laplacian
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
and of Euclidean space.
Nirenberg's first results on this problem were obtained in collaboration with
Basilis Gidas and
Wei-Ming Ni
Wei-Ming Ni (; born 23 December 1950) is a Taiwanese mathematician at the University of Minnesota and the Chinese University of Hong Kong, and the former director of the Center for PDE at the East China Normal University. He works in the field of ...
. 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 In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the ...
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 equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s 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
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
, 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
Ovidiu Vasile Savin (born January 1, 1977) is a Romanian-American mathematician who is active in the field of the partial differential equations.
Scientific activity
Savin received his Ph.D. in mathematics from the University of Texas at Austin ...
's Ph.D. thesis. They further applied their method to obtain qualitative phenomena on general unbounded domains, extending earlier works of
Maria Esteban and
Pierre-Louis Lions
Pierre-Louis Lions (; born 11 August 1956) is a French people, French mathematician. He is known for a number of contributions to the fields of partial differential equations and the calculus of variations. He was a recipient of the 1994 Fields Me ...
.
Functional inequalities
Nirenberg and
Emilio Gagliardo independently proved fundamental inequalities for
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s, now known as the
Gagliardo–Nirenberg–Sobolev inequality and the
Gagliardo–Nirenberg interpolation inequalities. 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. With
Luis Caffarelli
Luis Angel Caffarelli (born December 8, 1948) is an Argentine mathematician and luminary in the field of partial differential equations and their applications.
Career
Caffarelli was born and grew up in Buenos Aires. He obtained his Masters of S ...
and
Robert Kohn, Nirenberg would establish corresponding inequalities for certain weighted norms. 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
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a ...
'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, complex analysis, harmonic analysis, Fourier analysis, and partial differential equations.
Among other applications, it is particularly fundamental to
Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations.
Life
Moser's mother Ilse Strehl ...
's
Harnack inequality In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions ...
and subsequent work.
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 manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
s. Among other results, they were able to establish that smooth maps which are close in BMO norm have the same
topological degree In mathematics, topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution ...
, and hence that degree can be meaningfully defined for
mappings of vanishing mean oscillation (VMO).
Calculus of variations
In the setting of
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s,
Ky Fan
Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara.
Biography
Fan was born in Hangzhou, the capital of Zhejian ...
developed a
minimax theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality.
The first theorem in this sense is von Neumann's minimax theorem from 1928, which was c ...
with applications in
game theory
Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
. With
Haïm Brezis
Haïm Brezis (born 1 June 1944) is a French mathematician, who mainly works in functional analysis and partial differential equations.
Biography
Born in Riom-ès-Montagnes, Cantal, France. Brezis is the son of a Romanian immigrant father, wh ...
and
Guido Stampacchia
Guido Stampacchia (26 March 1922 – 27 April 1978) was an Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations..
Life and acade ...
, Nirenberg derived results extending both Fan's theory and Stampacchia's generalization of the
Lax-Milgram theorem Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or con ...
. Their work has applications to the subject of
variational inequalities In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities was initi ...
.
By adapting the
Dirichlet energy
In mathematics, the Dirichlet energy is a measure of how ''variable'' a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the ...
, it is standard to recognize solutions of certain
wave equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and s ...
s as
critical points of functionals. With Brezis and
Jean-Michel Coron
Jean-Michel Coron (born August 8, 1956) is a French people, French mathematician. He first studied at École Polytechnique, where he worked on his PhD thesis advised by Haïm Brezis. Since 1992, he has studied the control theory of partial differ ...
, 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 The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonstrates the existence of a saddle point. The th ...
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.
[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 In mathematical analysis, Ekeland's variational principle, discovered by Ivar Ekeland, is a theorem that asserts that there exist nearly optimal solutions to some optimization problems.
Ekeland's principle can be used when the lower level set of a ...
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 functions. Making use of an earlier regularity result of Brezis and
Tosio Kato
was a Japanese mathematician who worked with partial differential equations, mathematical physics and functional analysis.
Kato studied physics and received his undergraduate degree in 1941 at the Imperial University of Tokyo. After disruption o ...
, Brezis and Nirenberg ruled out such phenomena for a certain class of
Dirichlet-type functionals. Their work was later extended by Jesús García Azorero,
Juan Manfredi, and Ireneo Peral.
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
Thierry Aubin (6 May 1942 – 21 March 2009) was a French mathematician who worked at the Centre de Mathématiques de Jussieu, and was a leading expert on Riemannian geometry
and non-linear partial differential equations. His fundamental contrib ...
's work on the
Yamabe problem
The Yamabe problem refers to a conjecture in the mathematical field of differential geometry, which was resolved in the 1980s. It is a statement about the scalar curvature of Riemannian manifolds:
By computing a formula for how the scalar curvatur ...
. With Berestycki and Italo Capuzzo-Dolcetta, Nirenberg studied superlinear equations of Yamabe type, giving various existence and non-existence results.
Nonlinear functional analysis
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
:
to the spectral properties of the operator ''A''. Applications include the study of rather general parabolic and elliptic-parabolic problems.
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
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is giv ...
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 numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
(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
In mathematics, the Cauchy–Kovalevskaya theorem (also written as the Cauchy–Kowalevski theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A ...
, as a particular case of a theorem on solvability of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ...
s in families of
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
s. His work was later simplified by Takaaki Nishida and used in an analysis of the
Boltzmann equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
.
Geometric problems
Making use of his work on fully nonlinear elliptic equations, Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and
Minkowski problem
In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks for the construction of a strictly convex compact surface ''S'' whose Gaussian curvature is specified. More precisely, the input to the problem is a strictly posit ...
in the field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. The former asks for the existence of isometric embeddings of positively curved
Riemannian metrics on the two-dimensional sphere into three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, while the latter asks for closed surfaces in three-dimensional Euclidean space for which the
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...
prescribes the
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
. The key is that the "Darboux equation" from surface theory is of Monge−Ampère type, so that Nirenberg's regularity theory becomes useful in the
method of continuity.
John Nash's well-known
isometric embedding theorems, established soon afterwards, have no apparent relation to the Weyl problem, which deals simultaneously with high-regularity embeddings and low codimension.
Nirenberg's work on the Minkowski problem was extended to
Riemannian settings by
Aleksei Pogorelov
Aleksei Vasil'evich Pogorelov (russian: Алексе́й Васи́льевич Погоре́лов, ua, Олексі́й Васи́льович Погорє́лов; March 2, 1919 – December 17, 2002), was a Soviet and Ukrainian ...
. In higher dimensions, the Minkowski problem was resolved by
Shiu-Yuen Cheng
Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...
and
Shing-Tung Yau
Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau announced retirement from Harvard to become Chair Professor of mathem ...
.
Other approaches to the Minkowski problem have developed from Caffarelli, Nirenberg, and Spruck's fundamental contributions to the theory of nonlinear elliptic equations.
In one of his very few articles not centered on
analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
, Nirenberg and
Philip Hartman characterized the cylinders in Euclidean space as the only complete
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
s which are intrinsically flat. This can also be viewed as resolving a question on the isometric embedding of
flat manifold In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles o ...
s as hypersurfaces. Such questions and natural generalizations were later taken up by Cheng, Yau, and
Harold Rosenberg
Harold Rosenberg (February 2, 1906 – July 11, 1978) was an American writer, educator, philosopher and art critic. He coined the term Action Painting in 1952 for what was later to be known as abstract expressionism. Rosenberg is best known for ...
, among others.
[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
Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...
and
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
, Nirenberg and his doctoral student August Newlander proved what is now known as the
Newlander-Nirenberg theorem
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not compl ...
, which provides the precise algebraic condition under which an
almost complex structure
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
arises from a holomorphic coordinate atlas. The Newlander-Nirenberg theorem is now considered as a foundational result in
complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and c ...
, 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
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. Th ...
to the unit ball. Nirenberg and
Charles Loewner
Charles Loewner (29 May 1893 – 8 January 1968) was an American mathematician. His name was Karel Löwner in Czech and Karl Löwner in German.
Karl Loewner was born into a Jewish family in Lany, about 30 km from Prague, where his father Sig ...
studied the more general means of naturally assigning a complete Riemannian metric to bounded
open subset
In mathematics, open sets are a generalization of open intervals in the real line.
In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suff ...
s of
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. Geometric calculations show that solutions of certain semilinear
Yamabe-type equations 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.
Pseudo-differential operators
Joseph Kohn and Nirenberg introduced the notion of
pseudo-differential operator
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in ...
s. Nirenberg and
François Trèves investigated the famous
Lewy's example In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It shows that the analog of the Cauchy–Kovalevskaya theorem does ...
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
Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal ...
, 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.
Major publications
Books and surveys.
Articles.
References
External links
Homepage of Louis NirenbergSimons Foundation, Science Lives: Louis Nirenberg* Allyn Jackson
Interview with Louis Nirenberg.Notices Amer. Math. Soc. 49 (2002), no. 4, 441–449.
* YanYan Li
The work of Louis Nirenberg.Proceedings of the International Congress of Mathematicians. Volume I, 127–137, Hindustan Book Agency, New Delhi, 2010.
* Simon Donaldson
On the work of Louis Nirenberg.Notices Amer. Math. Soc. 58 (2011), no. 3, 469–472.
* Tristan Rivière
Exploring the unknown: the work of Louis Nirenberg on partial differential equations.Notices Amer. Math. Soc. 63 (2016), no. 2, 120–125.
Recent applications of Nirenberg's classical ideas.Communicated by Christina Sormani. Notices Amer. Math. Soc. 63 (2016), no. 2, 126–134.
* Martin Raussen and Christian Skau
Interview with Louis Nirenberg.Notices Amer. Math. Soc. 63 (2016), no. 2, 135–140.
* (Coordinated by Robert V. Kohn and Yanyan Li.
Louis Nirenberg (1925–2020).Notices Amer. Math. Soc. 68 (2021), no. 6, 959–979.
{{DEFAULTSORT:Nirenberg, Louis
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