利翁斯Pierre-Louis Lions

利翁斯是著名数学家雅克-路易·利翁斯和Andrée Olivier的儿子。他出生于法国普罗旺斯-阿尔卑斯-蓝色海岸大区滨海阿尔卑斯省的格拉斯，位于戛纳西北。这是他父亲的出生地，也是这个家庭视为家乡的城镇，尽管他出生时父亲是南锡大学的教授。让我们注意，格拉斯离德拉吉尼昂不远，而孔涅就出生在德拉吉尼昂，他比利翁斯获得约翰·查尔斯·菲尔兹奖章早12年获得菲尔兹奖。

利翁斯六岁时，他父亲成为巴黎的教授，全家住在那里。他先后就读于巴斯德中学和路易大帝中学，之后于1975年进入巴黎高等师范学院。利翁斯于1975年至1979年在巴黎高等师范学院学习。他的学位论文由H Brézis指导，提交给皮埃尔和玛丽·居里大学（1970年巴黎大学拆分为十三所独立大学时正式称为巴黎第六大学），并于1979年获得国家科学博士学位。1979年12月1日，获得博士学位后，利翁斯与Lila Laurenti结婚。他们有一个孩子Dorian。

1979年至1981年，利翁斯在巴黎的国家科学研究中心担任研究职位。然后，在1981年，他被任命为巴黎第九大学的教授。在仍担任这一职位期间，他于1995年被聘为国家科学研究中心的研究主任。自1992年起，他还担任巴黎综合理工学院的应用数学教授。

利翁斯在20世纪80年代和90年代对非线性偏微分方程理论做出了一些最重要的贡献。埃文斯在[2]中写道：-

他做出了真正根本性的发现，跨越了众多学科，纯数学与应用数学皆然，他的出版物如此众多且多样，以至于难以简单分类。请记住，非线性偏微分方程实际上并没有核心理论，也不可能有。偏微分方程的来源如此之多——物理的、概率的、几何的等等——以至于这门学科是不同子领域的联合体，每个子领域用完全不同的方法研究不同的非线性偏微分方程的不同现象。利翁斯的独特之处在于他难以置信地超越这些界限并解决整个领域中紧迫问题的能力。

引用的参考文献[2]、[3]和[4]描述了利翁斯工作的某些重要方面，这些工作导致他在1994年苏黎世国际数学家大会上获得菲尔兹奖章。[1]和[3]都强调的利翁斯工作的第一个领域是他关于非线性偏微分方程“粘性解”的工作。该方法最初由利翁斯在1983年与M G Crandall合作引入，他们研究了威廉·哈密顿-卡尔·古斯塔夫·雅各布·雅可比方程。此后，利翁斯和其他人将该方法应用于一大类偏微分方程，即所谓的“完全非线性二阶退化椭圆偏微分方程”。出现的问题在[2]中描述：-

……这类非线性偏微分方程在短时间后根本不存在光滑甚至$C^{1}$解。……因此唯一的选择是寻找某种“弱”解。这项工作实际上是弄清楚如何允许某些类型的“物理上正确”的奇点，以及如何禁止其他类型。……利翁斯和Crandall最终通过将注意力集中在粘性解上打开了这个问题，粘性解是根据某些不等式定义的，这些不等式在解的图形被光滑测试函数从一侧或另一侧接触到的任何地方都成立。

利翁斯的另一项同样创新的工作是他关于路德维希·玻尔兹曼方程和其他动理学方程的工作。路德维希·玻尔兹曼方程追踪碰撞粒子之间的相互作用，不是个别地而是通过密度。1989年，利翁斯在与DiPerma的合作中，首次给出了具有任意初始数据的严格解。

利翁斯的另一项重大贡献，体现在一系列重要的论文中，是对变分问题的贡献。S·R·S·瓦拉德汉在1994年苏黎世数学家大会上谈到利翁斯的工作[4]时说：-

有许多非线性偏微分方程是变分问题的莱昂哈德·欧拉方程。用变分方法求解此类方程的第一步是证明极值被达到。这需要某种强制性或紧性。如果要最小化的量有一个涉及导数的类似“能量”的项，那么沿着最小化序列就有对局部正则性的控制。

利翁斯的巧妙想法是引入“集中紧性”技巧，它着眼于能量集中，从而避免在考察极小化序列时因缺乏紧性而出现的问题。他引入了某些测度来处理集中现象。

利翁斯因其对数学的杰出贡献而获得许多奖项。他是法国科学院的成员，并获得了科学院颁发的奖项，1986年的Doistau-Blutet基金会奖和1992年的安德烈-马里·安培奖。他还获得了1987年的IBM奖和1991年的Philip Morris奖。

除了巴黎科学院之外，利翁斯还被选为那不勒斯科学院和欧洲科学院的成员。他还是荣誉军团骑士。他被苏格兰爱丁堡的赫瑞瓦特大学授予荣誉博士学位。他是全球约25种期刊的编委会成员。

最后让我们注意到，利翁斯被授予Thomson奖（2004年），并代表他的团队被授予Europlace金融研究所奖（2003年）。

利翁斯将他的爱好列为电影和阅读，他最喜欢的运动是橄榄球和游泳。

Pierre-Louis Lions is the son of the famous mathematician Jacques-Louis Lions and Andrée Olivier. He was born in Grasse, Alpes-Maritimes in the Provence- Alpes- Côte- d'Azur region of France, northwest of Cannes. It was the birthplace of his father and the town the family considered home, although at the time of his birth his father was a professor at the University of Nancy. Let us note that Grasse is not very far from Draguignan where Alain Connes, who won a Fields Medal 12 years before Lions won his Fields Medal, was born.

When Pierre-Louis was six years old his father became a professor in Paris and the family lived there. He attended the Lycée Pasteur and then the Lycée Louis-le-Grand before entering the École Normale Supérieure in 1975. Lions studied at the École Normale Supérieure from 1975 to 1979. His thesis, supervised by H Brézis, was presented to the University of Pierre and Marie Curie (formally Paris VI when the University of Paris was split into thirteen separate universities in 1970) and in 1979 he received his Doctorat d'Etat es sciences. On 1 December 1979, after he received his doctorate, Lions married Lila Laurenti. They have one child Dorian.

From 1979 to 1981 Lions held a research post at the Centre National de la Recherche Scientifique in Paris. Then, in 1981, he was appointed professor at the University of Paris-Dauphine. While still holding this post he was attached to the Centre National de la Recherche Scientifique as Director of Research in 1995. He has also held the position of Professor of Applied Mathematics at the École Polytechnique from 1992.

Lions has made some of the most important contributions to the theory of nonlinear partial differential equations through the 1980s and 1990s. Evans, in [2], writes:-

He has made truly fundamental discoveries cutting across many disciplines, pure and applied, and his publications are so numerous and varied as to defy easy classification. Keep in mind that there is in truth no central core theory of nonlinear partial differential equations, nor can there be. The sources of partial differential equations are so many - physical, probalistic, geometric etc. - that the subject is a confederation of diverse subareas, each studying different phenomena for different nonlinear partial differential equation by utterly different methods. Pierre-Louis Lions is unique in his unbelievable ability to transcend these boundaries and to solve pressing problems throughout the field.

The references quoted [2], [3] and [4] decribe some important aspects of Lions work which led to the award of a Fields Medal at the International Congress of Mathematicians in Zürich in 1994. The first area of Lions work that is highlighted by both [1] and [3] is his work on "viscosity solutions" for nonlinear partial differential equations. The method was first introduced by Lions in joint work with M G Crandall in 1983 in which they studied Hamilton-Jacobi equations. Lions and others have since applied the method to a wide class of partial differential equations, the so-called "fully nonlinear second order degenerate elliptic partial differential equations." The problem that arises is decribed in [2]:-

... such nonlinear partial differential equation simply do not have smooth or even $C^{1}$ solutions existing after short times. ... The only option is therefore to search for some kind of "weak" solution. This undertaking is in effect to figure out how to allow for certain kinds of "physically correct" singularities and how to forbid others. ... Lions and Crandall at last broke open the problem by focusing attention on viscosity solutions, which are defined in terms of certain inequalities holding wherever the graph of the solution is touched on one side or the other by a smooth test function.

Another equally innovative piece of work by Lions was his work on the Boltzmann equation and other kinetic equations. The Boltzmann equation keeps track of interactions between colliding particles, not individually but in terms of a density. In 1989 Lions, in joint work with DiPerma, was the first to give a rigorous solution with arbitrary initial data.

Another major contribution by Lions, in a long series of important papers, is to variational problems. Varadhan, speaking at the Congress of Mathematicians in Zürich in 1994 about Lions' work [4], said:-

There are many nonlinear PDEs that are Euler equations for variational problems. The first step in solving such equations by the variational method is to show that the extremum is attained. This requires some coercivity or compactness. If the quantity to be minimised has an "energy"-like term involving derivatives, then one has control on local regularity along a minimising sequence.

Lions's clever idea was to introduce "concentration compactness" techniques which look at energy concentrations and so avoid problems which occur when examining the minimising sequences without compactness. He introduced certain measures to handle the concentrations.

Lions has received many awards for his outstanding contributions to mathematics. He is a member of the French Academy of Sciences and he was awarded prizes by the Academy, the Doistau-Blutet Foundation Prize in 1986 and the Ampère Prize in 1992. He also received the IBM Prize in 1987 and the Philip Morris Prize in 1991.

In addition to the Paris Academy, Lions has been elected a member of the Naples Academy and the European Academy. He is also Chevalier of the Légion d'Honneur. He has been awarded an honorary doctorate from Heriot-Watt University in Edinburgh, Scotland. He is on the editorial board of around 25 journals world-wide.

Finally let us note that Lions was awarded the Prix Thomson (2004) and, on behalf of his team, the Prix Institut de Finance Europlace (2003).

Lions lists his hobbies as cinema and reading, and his favourite sports as rugby and swimming.