路易·尼伦伯格Louis Nirenberg

路易·尼伦伯格的父亲是一位希伯来语教师。尼伦伯格上了一所希伯来学校，但他不想学希伯来语。他的父亲试图教他，但遇到抵触，认为朋友给他上私人课程可能更成功。这位朋友热爱数学谜题，因此大部分课程都花在解谜上，而不是学习希伯来语。尼伦伯格在蒙特利尔的Baron Byng高中就读，那里有优秀的教师。在这个阶段，主要因为一位杰出的物理教师，他认为自己想成为一名物理学家。从该校毕业后，他进入麦吉尔大学，主修数学和物理。他于1945年从麦吉尔大学毕业，获得学士学位。

毕业后，尼伦伯格认为他会继续攻读理论物理的研究生。然而，毕业后他在蒙特利尔的加拿大国家研究委员会找了一份暑期工作。当时他们的研究是关于原子弹项目的，那里的一位科学家是Ernst Courant，即理查·科朗特的长子。Ernst Courant娶了一位来自蒙特利尔的女孩，尼伦伯格认识她，所以他问她，下次拜访他时，能否向理查·科朗特请教学习理论物理的最佳地点。得到的建议是，尼伦伯格应该先在纽约大学攻读数学硕士学位，然后再去学习物理。尼伦伯格去纽约与理查·科朗特和卡特·佛莱德里克斯面谈，他们给了他一个助教职位。

1947年获得纽约大学硕士学位后，尼伦伯格留在那里攻读博士学位。他的官方导师是Jim Stoker，但他深受卡特·佛莱德里克斯的影响。他于1949年因学位论文The Determination of a Closed Convex Surface Having Given Line Elements获得博士学位。在这篇论文中，他考察了一个赫尔曼·外尔在1916年左右首次提出并部分解决的问题：

给定单位球面上的一个黎曼度量，具有正的卡尔·弗里德里希·高斯曲率，你能将这个2-球面等距嵌入到3维空间中作为一个凸曲面吗？

他利用Charles Morrey的想法对这个问题给出了肯定的回答。1953年，他在论文The Weyl and Minkowski problems in differential geometry in the large中发表了他的学位论文的结果。同样在1953年，即他的出版物首次出现的那一年，他又发表了三篇论文：A strong maximum principle for parabolic equations；A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type；和On nonlinear elliptic partial differential equations and Hölder continuity。

1949年完成博士学位后，尼伦伯格被任命为纽约大学的研究助理，成为数学科学与力学研究所（1964年更名为理查·科朗特研究所）的早期成员之一。从研究助理晋升为副研究员后，他于1951年被聘为助理教授。1951-52年他在欧洲度过，访问了苏黎世和哥廷根。在苏黎世，他参加了海因茨·霍普夫关于几何的讲座课程、巴特尔·伦德特·范德瓦尔登关于波恩哈德·黎曼曲面理论的课程，以及罗尔夫·内万林纳和沃尔夫冈·泡利的讲座。他回到纽约，在1957年成为理查·科朗特研究所的正教授之前被提升为副教授。他在1958-60年期间是斯隆研究员，并在1966-67年和1975-76年再次成为古根海姆研究员。

我们通过引用 尼伦伯格 于1994年从 美国数学会 获得的 凯瑟琳·斯蒂尔 终身成就奖的颁奖词来概述他的数学贡献 [1]：-

尼伦伯格是分析和所有领域中获取和应用先验估计的艺术与科学的大师。一个小巧的瑰宝是有用的Garliardo-尼伦伯格不等式集。一个高峰是他与[Shmuel] Agmon和[Avron] Douglis关于一般线性椭圆系统的先验估计的联合研究，这是分析中被引用最广泛的结果之一。另一个是他与弗瑞兹·约翰关于有界平均振荡函数的基本论文，这对查尔斯·费夫曼后来关于这个函数空间的工作至关重要。尼伦伯格一直是许多重大发展的中心。他与学生Newlander关于几乎复结构的定理已成为经典。在一篇基于阿尔伯托·考尔德伦和[Antoni] Zygmund早期估计的论文中，他和[Joseph] Kohn引入了伪微分算子的概念，这有助于产生大量后续工作。他与[François] Trèves的研究对一般线性偏微分方程的可解性做出了重要贡献。其他一些亮点包括他与[弗洛伦斯·南丁格尔·大卫] Kinderlehrer和[Joel] Spuck关于自由边界问题正则性的研究，与路易斯·卡法雷利和Spuck关于加斯帕尔·蒙日-安德烈-马里·安培型方程光滑解的存在性，以及与路易斯·卡法雷利和[Robert] Kohn关于克洛德-路易·纳维-乔治·加布里埃尔·斯托克斯方程的奇异集。他与[Basilis] Gidas和[Wei Ming] Ni以及后来与[Henri] Berestycki使用移动平面法对非线性椭圆方程对称解的研究，是最大值原理的巧妙应用。

在这段引文开头附近提到的不等式，在尼伦伯格的生活中占有特殊地位。他被引述说[5]：-

卡特·佛莱德里克斯 非常喜爱不等式，这对我影响很大。他的观点是不等式比等式更有趣。

他又说[4]：-

我喜欢不等式。所以如果有人向我展示一个新不等式，我会说：“哦，那很美，让我想想，”我可能会有一些与之相关的想法。

尼伦伯格 的众多出版物（MathSciNet 列出 185 篇）中只有一本书。这就是 Topics in non-linear functional analysis（1974），它是 Ralph A Artino 对 尼伦伯格 于 1973-74 年在 理查·科朗特 研究所讲授的一门课程的记录[2]：-

该课程面向在线性算子方面有良好基础、对 微分几何 有一定了解但拓扑学知识有限的学生。前两章致力于拓扑度方法，首先在有限维空间中，然后在 斯特凡·巴拿赫 空间中。接下来关于分岔理论的一章之后是由 J Ize 合写的一章，介绍了进一步的拓扑方法，如配边和上同伦。第五章简要涉猎单调算子理论。

Jerrold Marsden 对这本书的评论开头如下：-

对于非线性分析的学生和研究者来说，这本讲义是当前可获得的关于该主题最有用的入门书。它简短、简洁、切中要害，证明异常优雅，总是带有几何风味，且是现有最好的。这些讲义的大部分风味集中在度理论的使用上。在大约五十页的篇幅内引入度的概念并发展其基本性质，这本身就是一个胜利。这在第三章和第四章中用于讨论分支理论（亮点是Rabinowitz全局分支定理的完整证明）和非线性偏微分方程的求解（亮点是Landesman和Lazer的全局定理）。

这本关于非线性问题的书的修订重印版于2001年出版。我们应该注意到尼伦伯格对非线性问题的特殊热爱，并引用他关于这个主题的两段话。以下是第一段[5]：-

非线性问题的大多数结果仍然是通过线性问题获得的，即尽管问题是非线性的，但不是因为非线性。

第二段引文与他所描述的某位作者的结果有关[5]：-

方程的非线性特征被以本质的方式使用，事实上他获得结果是因为非线性，而不是尽管有非线性。

尼伦伯格在其杰出的职业生涯中获得了许多奖项和荣誉。他于1959年获得了美国数学会的马克希莫·博谢奖：-

……因他在偏微分方程方面的工作。

1962年8月，他在斯德哥尔摩举行的国际数学家大会上作全会报告，演讲题目为Some Aspects of Linear and Nonlinear Partial Differential Equations。在这次演讲中，他综述了非线性边值问题，特别是椭圆型问题，在存在性和正则性理论方面的一些近期进展。1983年，他还在苏格兰阿伯丁举行的英国数学讨论会上作全会报告，演讲题目为Comments on non-linear problems。

尼伦伯格曾以多种身份为美国数学会服务。他在1963-65年间担任理事会成员，1974年作了美国数学会讨论会讲座，并于1976-77年担任该学会副主席。1982年9月29日星期三，在瑞典皇家科学院举行的颁奖仪式上，克拉福德奖被授予尼伦伯格。他与弗拉基米尔·阿诺尔德因在非线性微分方程领域的成就而共同获得了35万瑞典克朗的奖金。他们是首批获得克拉福德奖的数学家。新闻稿包含以下内容：-

艾萨克·牛顿的思想被用来描述物理学和几何学中许多不同的系统，这些系统可能同时依赖于多个变量。相应的方程于是被称为偏微分方程，而其中最有趣的又是非线性的。尼伦伯格处理了大量这类问题。几何学方面的一个例子是寻找具有给定曲率的曲面，物理学方面则有粘性流体方程的研究以及关于自由流线存在性的研究。尼伦伯格的工作涵盖各种各样的问题，对该领域的发展具有根本的重要性。

他于1987年获得了 加拿大数学会 的 Jeffery-Williams 奖。我们已在上文提到他于1994年从 美国数学会 获得的 斯蒂尔 终身成就奖。最近，他于1995年10月18日获得了国家科学奖章。在这一最负盛名的奖项颁发时，理查·科朗特 研究所所长 大卫 W McLaughlin 说 [3]：-

尼伦伯格的影响并不限于他对此学科做出的许多原创性和基础性贡献。他不仅在世界范围内数学分析的发展中发挥了重要作用，而且对年轻数学家的成长也产生了显著影响，这一点从他指导了四十多名博士生即可看出。作为理查·科朗特的前任所长，他展现了对数学科学界的领导力和远见。他写作、演讲以及众多阐述性文章的清晰性，继续激励着一代又一代的数学家。

在文章3中，尼伦伯格的两位合作者路易斯·卡法雷利和Joseph Kohn描述了他的数学贡献。路易斯·卡法雷利写道[3]：-

尼伦伯格的工作极大地影响了与偏微分方程以某种方式相关联的所有数学领域：实分析与复分析、变分法、微分几何、连续介质与流体力学。

路易斯·卡法雷利提到了尼伦伯格在偏微分方程方面的兴趣领域：二阶椭圆方程的正则性与可解性n；赫尔曼·闵可夫斯基问题与完全非线性方程；自由边界问题的高阶正则性理论；以及不变方程解的对称性质。

Joseph Kohn描述了尼伦伯格对复分析的贡献。他以对尼伦伯格贡献的这一概述开始[3]：-

尼伦伯格因其对数学的许多重要研究贡献、精彩的讲座和清晰的阐述性写作而享誉世界。他的兴趣范围非常广泛：微分方程、调和分析、微分几何、泛函分析、复分析等。……尼伦伯格的职业生涯一直是一种激励；他的众多学生、合作者和同事从他那里学到了很多。除了数学之外，尼伦伯格还教会了我们所有人享受旅行、电影和美食。对尼伦伯格的欣赏还必须包括他始终如一的幽默感。他的幽默无法抑制，因此有时会出现在印刷页面上。

尼伦伯格已当选为国家科学院、美国哲学学会、American Academy of Arts and Sciences、Accademia dei Lincei、法国Académie des Sciences、地中海科学院、伦巴第研究所、科学与文学科学院以及乌克兰科学院的成员。他获得了麦吉尔大学（1986年）、比萨大学（1990年）和巴黎第九大学巴黎-多芬纳（1990年）的荣誉学位。

最后我们提到，1990年9月在意大利特伦托举行了一次献给尼伦伯格的会议，以纪念他的65岁生日。1996年6月在威尼斯举行了一次会议，1996年7月在佛罗伦萨又举行了一次会议，都是为了庆祝尼伦伯格的70岁生日。2000年9月在新竹举行了一次会议，以庆祝尼伦伯格的75岁生日；2006年6月在托莱多的卡斯蒂利亚-拉曼恰大学又举行了一次会议，以庆祝尼伦伯格的80岁生日。

尼伦伯格和约翰·福布斯·纳什于2015年获得尼尔斯·阿贝尔奖，获奖理由为：-

……对非线性偏微分方程理论及其在几何分析中的应用做出了引人注目且具有开创性的贡献。

让我们以尼伦伯格[4]的一句话作为结束：-

我和Philip Hartman合写过一篇论文，虽然浅显，但做起来非常有趣。这正是我想让那些对数学一无所知的人明白的——数学多么有趣！数学的奇妙之处之一在于，你走到世界某个地方，遇到其他数学家，就像一个大家庭。这个大家庭是一种美妙的喜悦。

Louis Nirenberg's father was a teacher of Hebrew. Louis went to a Hebrew school but he did not want to learn Hebrew. His father tried to teach him but, meeting resistance, thought that a friend might have more success giving him private lessons. This friend loved mathematical puzzles and so most of the lessons were spent solving puzzles rather than learning Hebrew. Louis attended Baron Byng High School in Montreal where he had excellent teachers. At this stage, primarily because of an outstanding physics teacher, he thought he would like to become a physicist. After graduating from the school, he entered McGill University where he majored in both mathematics and physics. He graduated from McGill with a B.S. in 1945.

After he graduated Nirenberg thought that he would carry on to graduate studies in theoretical physics. However, after graduating he took a summer job at the National Research Council of Canada in Montreal. At that time their research was on the atomic bomb project and one of the scientists there was Ernst Courant, Richard Courant's eldest son. Ernst Courant had married a girl from Montreal who Nirenberg knew, so he asked her if, when she next visited him, she could ask Richard Courant for advice on the best place to study theoretical physics. The advice came back that Nirenberg should first do a Master's Degree in mathematics at New York University, and then go on to study physics. Nirenberg went for an interview with Courant and Friedrichs in New York and they offered him an assistantship.

After the award of a Master's Degree by New York University in 1947, Nirenberg remained there working for his doctorate. His official supervisor was Jim Stoker but he was greatly influenced by Kurt Friedrichs. He was awarded a Ph.D. in 1949 for his thesis The Determination of a Closed Convex Surface Having Given Line Elements. In this he examined a question first asked, and partially solved, by Hermann Weyl around 1916:

Given a Riemannian metric on the unit sphere, with positive Gauss curvature, can you embed this 2-sphere isometrically into 3-space as a convex surface?

He used ideas due to Charles Morrey in giving a positive answer to this question. He published the results of his thesis in 1953 in the paper The Weyl and Minkowski problems in differential geometry in the large. Also in 1953, the first year in which his publications appear, he published three further papers: A strong maximum principle for parabolic equations; A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic-hyperbolic type; and On nonlinear elliptic partial differential equations and Hölder continuity.

After completing his doctorate in 1949, Nirenberg was appointed as a Research Assistant at New York University, becoming one of the early members of the Institute of Mathematical Sciences and Mechanics (it was renamed the Courant Institute in 1964). After being promoted from Research Assistant to Research Associate, he was made an Assistant Professor in 1951. He spent 1951-52 in Europe, visiting Zürich and Göttingen. In Zürich he attended a lecture course by Heinz Hopf on geometry, a course by Bartel van der Waerden on Riemann surface theory, and lectures by Rolf Nevanlinna and Wolfgang Pauli. He returned to New York and was promoted to Associate Professor before becoming a full professor at the Courant Institute in 1957. He was a Sloan Fellow during 1958-60, and a Guggenheim Fellow in 1966-67 and again in 1975-76.

We give an overview of Nirenberg's mathematical contributions by quoting from the citation for the Steele Prize for Lifetime Achievement which he received from the American Mathematical Society in 1994 [1]:-

Nirenberg is a master of the art and science of obtaining and applying a priori estimates in all fields of analysis. A minor such gem is the useful set of Garliardo-Nirenberg inequalities. A high point is his joint research with [Shmuel] Agmon and [Avron] Douglis on a priori estimates for general linear elliptic systems, one of the most widely quoted results in analysis. Another is his fundamental paper with Fritz John on functions of bounded mean oscillation which was crucial for the later work of [Charles] Fefferman on this function space. Nirenberg has been the centre of many major developments. His theorem with his student, Newlander, on almost complex structures has become a classic. In a paper building on earlier estimates of [Alberto] Calderón and [Antoni] Zygmund, he and [Joseph] Kohn introduced the notion of a pseudo-differential operator which helped to generate an enormous amount of later work. His research with [François] Trèves was an important contribution to the solvability of general linear PDEs. Some other highlights are his research on the regularity of free boundary problems with [David] Kinderlehrer and [Joel] Spuck, existence of smooth solutions of equations of Monge-Ampère type with [Luis] Caffarelli and Spuck, and singular sets for the Navier-Stokes equations with Caffarelli and [Robert] Kohn. His study of symmetric solutions of non-linear elliptic equations using moving plane methods with [Basilis] Gidas and [Wei Ming] Ni and later with [Henri] Berestycki, is an ingenious application of the maximum principle.

Inequalities, mentioned near the beginning of this quote, have a special part in Nirenberg's life. He is quoted as saying [5]:-

Friedrichs was a great lover of inequalities and that affected me very much. The point of view was that the inequalities are more interesting than the equalities.

Again he said [4]:-

I love inequalities. So if somebody shows me a new inequality, I say: "Oh, that's beautiful, let me think about it," and I may have some ideas connected to it.

The many publications by Nirenberg (MathSciNet lists 185) include only one book. This is Topics in non-linear functional analysis (1974) which is a record by Ralph A Artino of a lecture course given by Nirenberg at the Courant Institute in 1973-74 [2]:-

The lecture course was aimed at students with a good grounding in linear operators and some familiarity with differential geometry, but limited knowledge of topology. The first two chapters are devoted to topological degree approaches, first in finite-dimensional spaces and then in Banach space. The next chapter on bifurcation theory is followed by one co-written by J Ize, which introduces further topological methods such as cobordism and cohomotopy. Chapter five is devoted to a brief foray into the theory of monotone operators.

Jerrold Marsden begins a review of the book as follows:-

For students and researchers in nonlinear analysis this volume of lecture notes is the most useful introduction to the subject currently available. It is short, concise and to the point, and the proofs are unusually elegant, always with a geometric flavour and the best available. Much of the flavour of the notes centres on the use of degree theory. The introduction of the notion of degree and the development of its basic properties in the span of some fifty pages is a triumph in itself. This is used in Chapters III and IV in the discussion of bifurcation theory (the highlight being a complete proof of Rabinowitz' global bifurcation theorem) and the solution of nonlinear partial differential equations (the highlight being the global theorem of Landesman and Lazer).

A revised reprint of this book on nonlinear problems was published in 2001. We should note Nirenberg's special love of nonlinear problems and give two quotes from him on this topic. Here is the first [5]:-

Most results for nonlinear problems are still obtained via linear ones, i.e. despite the fact that the problems are nonlinear not because of it.

The second quote related to an author's results he was describing [5]:-

The nonlinear character of the equations is used in an essential way, indeed he obtains results because of the nonlinearity not despite it.

Nirenberg has been awarded many prizes and given many honours in a distinguished career. He received the Bôcher Prize from the American Mathematical Society in 1959:-

... for his work in partial differential equations.

He was a plenary speaker at the International Congress of Mathematicians held in Stockholm in August 1962, giving the lecture Some Aspects of Linear and Nonlinear Partial Differential Equations. In this lecture he gave a survey of some recent developments in existence and regularity theory for nonlinear boundary value problems, especially elliptic problems. He was also a plenary speaker at the British Mathematical Colloquium in Aberdeen, Scotland, in 1983 when he gave the lecture Comments on non-linear problems.

Nirenberg has served the American Mathematical Society in several roles. He was a member of the Council during 1963-65, gave the American Mathematical Society Colloquium Lectures in 1974, and was Vice-President of the Society in 1976-77. The Crafoord Prize was awarded to Nirenberg at a ceremony at the Royal Swedish Academy of Sciences on Wednesday, 29 September 1982. He shared the Prize of 350000 Swedish crowns with Vladimir Igorevich Arnold for their achievements in the field of non-linear differential equations. They were the first mathematicians to receive the Crafoord Prize. The Press Release contains the following:-

Newton's ideas were used to describe many different systems in physics and in geometry, which may depend on several variables simultaneously. The equations are then called partial differential equations and again the most interesting ones are non-linear. Nirenberg has treated a great number of problems of this nature. As an example from geometry one can mention the problem to find a surface with given curvature and from physics studies of the equations for viscose fluids and concerning existence of free streamlines. Nirenberg's work ranges over a variety of problems and has been of basic importance for the development of the field.

He received the Jeffery-Williams Prize of the Canadian Mathematical Society in 1987. We have already mentioned above the Steele Prize for Lifetime Achievement which he received from the American Mathematical Society in 1994. Most recently, he received the National Medal of Science on 18 October 1995. At the time of this most prestigious award, David W McLaughlin, the director of the Courant Institute, said [3]:-

Nirenberg's influence is not limited to the many original and fundamental contributions he has made to the subject. He has not only played a major role in the development of mathematical analysis worldwide but has had significant influence on the development of young mathematicians, as indicated by his direction of over forty Ph.D. students. As a past director of Courant, he has demonstrated leadership and vision for the mathematical sciences community. The clarity of his writing, his lectures, and numerous expository articles continue to inspire generations of mathematicians.

In the article [3] two of Nirenberg's collaborators, Luis Caffarelli and Joseph Kohn, describe his mathematical contributions. Caffarelli writes [3]:-

The work of Louis Nirenberg has enormously influenced all areas of mathematics linked one way or another with partial differential equations: real and complex analysis, calculus of variations, differential geometry, continuum and fluid mechanics.

Caffarelli mentions Nirenberg's areas of interest in partial differential equations: Regularity and solvability of elliptic equations of order 2n; the Minkowski problem and fully nonlinear equations; the theory of higher regularity for free boundary problems; and symmetry properties of solutions to invariant equations.

Joseph Kohn describes Nirenberg's contributions to complex analysis. He begins with this overview of Nirenberg's contributions [3]:-

Louis Nirenberg is recognized throughout the world for his many important research contributions to mathematics, his marvellous lectures, and his lucid expository writing. His range of interest is very broad: differential equations, harmonic analysis, differential geometry, functional analysis, complex analysis, etc. ... Nirenberg's career has been an inspiration; his numerous students, collaborators, and colleagues have learned a great deal from him. Aside from mathematics, Nirenberg has taught all of us the enjoyment of travel, movies, and gastronomy. An appreciation of Nirenberg also must include his ever-present sense of humour. His humour is irrepressible, so that on occasion it makes its way to the printed page.

Nirenberg has been elected to the National Academy of Sciences, the American Philosophical Society, the American Academy of Arts and Sciences, the Accademia dei Lincei, the French Académie des Sciences, the Accademia Mediterranea della Scienze, the Istituto Lombardo, the Accademia Scienze e Lettere, and the Ukrainian Academy of Sciences. He has received honorary degree from McGill University (1986), the University of Pisa (1990), and Université de Paris IX Paris-Dauphine (1990).

Finally we mention that a conference dedicated to Louis Nirenberg was held in Trento, Italy in September 1990 to mark his 65th birthday. A conference was held in Venice in June 1996 and another was held in Florence in July 1996, both to honour Nirenberg's 70th birthday. A conference was held in Hsinchu in September 2000 to honour Nirenberg's 75th birthday, and another conference was held at the Universidad de Castilla-La Mancha, Toledo in June 2006 in honour of Nirenberg's 80th birthday.

Nirenberg and John Nash were awarded the Abel prize in 2015 for:-

... striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.

Let us end with a quote from Nirenberg [4]:-

I wrote one paper with Philip Hartman that was elementary but enormous fun to do. That's the thing I try to get across to people who don't know anything about mathematics, what fun it is! One of the wonders of mathematics is you go somewhere in the world and you meet other mathematicians, and it's like one big family. This large family is a wonderful joy.