Proceedings of the International Congress of Mathematicians: August 3–11, 1994 Zürich, Switzerland

Abstract

Louis Nirenberg is one of the most outstanding analysts of the twentieth century.Formorethanhalfacentury,hehasbeenaworldleaderinpartialdifferential equations – a master of inequalities and regularity theory – with fundamental contribution in geometry, complex analysis, and fluid dynamics. Nirenberg is a household name in these fields. In addition to the depth and its basic importance, his work also has enormous influence on others. In each of the last 10 years, top 15 cited papers in mathematics include at least 2 of Nirenberg ′ s, according to the MathSciNet. Working with others has been an essential part of Nirenberg ′ s research — more than 90 % of his research are joint works. 1. Geometry In 1953 Nirenberg published 4 papers. The first two papers ([1] and [2]) solved two long standing open problems, the Weyl problem and the Minkowski problem, in differential geometry, and, in partial differential equations, gave basic estimates to solutions of nonlinear second order elliptic equations in dimension two. His solution of the Weyl and Minkowski problem was a pioneering work in the study of geometry problems using nonlinear partial differential equations, and a milestone in global geometry. These were established in his Ph.D. thesis in 1949. He was slow in rewriting the thesis for publication. His thesis adviser was J.J. Stoker, himself a student of Heinz Hopf. The Weyl problem, raised by H. Weyl in 1916, is the following: Given a smooth metric g of positive Gauss curvature on the sphere S 2, is there an embedding X: S 2 → R 3 such that the metric induced on S 2 by this embedding is g? Such an embedding (S 2,g) → R 3 is called isometric, and satisfies the following system of nonlinear partial differential equations: ∇iX ·∇jX = gij. Such an isometric embedding, if it exists, is unique up to rigid motion. H. Lewy proved in 1938 the existence part under the assumption that the metric