# Leray, Jean

# LERAY, JEAN

(*b*. Chantenay [now in Nantes], France, 7 November 1906; *d*. La Baule, France, 10 November 1998),

*mathematics, particularly algebraic topology and elasticity*.

Leray made fundamental contributions to fluid mechanics, in particular the existence of classical and weak solutions of Navier-Stokes equations, to partial differential equations, especially linear hyperbolic or analytic equations, and nonlinear elliptic Dirichlet problems, nonlinear functional analysis, with Leray-Schauder degree and monotone-like operators; algebraic topology, introducing sheaves and spectral sequences; elasticity; functions of several complex variables, with the Cauchy-Fantappié-Leray formula; and Lagrangian analysis.

**Early Life and Career** . Jean Leray was the son of two teachers, Francis Leray and Baptistine Pineau. After graduating at the École Normale Supérieure in Paris in 1929, Leray defended his PhD thesis, “Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique” (A study of various nonlinear integral equations and of some hydrodynamical problems), at the Faculté des sciences de Paris in 1933. His advisor was Henri Villat. One year before, Leray had married Marguerite Trumier, with whom he had three children, Jean-Claude, Françoise, and Denis. When World War II started, Jean Leray was a professor at the Faculté des sciences of Nancy. He served as a reserve officer in the French army but was captured by the Germans in June 1940 and sent to the prison camp Oflag XVIIA at Edel-bach, near Austerlitz (in what was later Czechoslovakia). There, he organized and led a university until the liberation of the camp in May 1945. After two years at the Faculté des sciences de Paris, Leray became a professor at the Collège de France in Paris (holding the chair of differential and functional equations) until his retirement in 1978.

A few years before his death, Leray left Sceaux for la Baule, near Nantes. He became a Fellow of the French Académie des sciences (mechanics section) in 1953, and later of most of the prestigious science academies in Europe and the United States. Besides receiving several prizes from the French Académie des sciences, Leray was awarded the prestigious Malaxa (1938), Feltrinelli (1971), and Wolf (1979) prizes.

**Leray’s Personality** . Leray’s mathematical work extended over an exceptionally long period; more than sixty years separated his first and his last papers. His contributions are highly original and cover an unusually wide spectrum of mathematics, from algebraic topology to elasticity. Leray’s independence of mind prevented him from allying himself with a political party, school, group, or ideology. He was among the few important mathematicians to react publicly against some excesses in the use of “modern mathematics” in French high school programs. His incisive sarcasm is reminiscent of Voltaire and Henri Poincaré, and the severe elegance of his language follows from a constant care for concision. Leray’s lectures, austere and deprived of embellishments, required a great deal from his listeners.

**Fluid Mechanics** . Leray’s work can for the most part be described in a chronological way, because after moving on to a new subject, he almost never came back to an earlier interest. In his PhD thesis, Leray successfully combined the Lyapunov-Schmidt reduction method with the ArzelàAscoli theorem and the existence of a priori bounds to obtain global existence results for nonlinear integral equations and stationary solutions of the equations of hydrodynamics through analytical continuation. The corresponding Navier-Stokes evolution equations are masterly treated in two memoirs published in 1934, introducing for the first time fundamental concepts such as weak solutions (called turbulent solutions by Leray) and the Sobolev space *H*^{1}(**R** ^{3}). For the Cauchy problem, he proved the existence of at least one global weak solution, which is regular and unique near the initial time. The global uniqueness and regularity is still open.

In 1933 Leray discovered, at a meeting in Paris with the Polish mathematician Julius Schauder, the topological techniques required by his thesis. The consequence was a joint paper worked out in two weeks in the city’s Jardin du Luxembourg (Luxembourg Garden). Topological degree theory in infinite-dimensional Banach spaces was born, as well as the global theory of nonlinear elliptic partial differential equations. The Leray-Schauder degree and the continuation theorem have inspired the entire development of nonlinear functional analysis. Leray-Schauder’s fixed point theorem, in its simplest version, ensures the existence of at least one fixed point for a completely continuous mapping *T* on a Banach space, when the set of possible fixed points of *ΛT* is a priori bounded independently of *Λ*. In the remaining years before World War II, Leray initiated algebraic topology in infinite dimensional Banach spaces through his product formula for degree, and he also successfully applied the new continuation method to the theory of wakes and bows and to fully nonlinear Dirichlet problems.

Afraid that his expertise in fluid mechanics could lead the Germans to force him into collaboration with their war effort, Leray concentrated his teaching in the Oflag on algebraic topology, reconstructed through an original approach. In order to avoid finite-dimensional approximations and the restriction to linear spaces, Leray tried to determine the relation between the cohomology of the source, the target, and the fiber of a continuous mapping. To achieve this aim, he introduced the seminal concept of “sheaf” on a topological space (a general tool to go from local to global results) and the powerful method of spectral sequences. In his hands and those of other mathematicians, such as Henri Cartan, Jean-Pierre Serre, Armand Borel, Jean-Louis Koszul, and Alexandre Grothendieck, those tools not only revolutionized algebraic topology but also the theory of functions of several complex variables; homological algebra; algebraic geometry; and more recently, algebraic analysis. Leray applied his new ideas to the cohomology of closed continuous maps, fiber spaces, and Lie groups in what came to be called the Leray-Hirsch theorem.

**Hyperbolic Partial Differential Equations** . In the early 1950s, Leray shifted his interest to linear hyperbolic partial differential equations of arbitrary order, treated by Jacques Hadamard and Julius Schauder in the second order case. When the coefficients were constants, he applied Schwartz’s distributions and algebraic geometry to construct the elementary solutions in what became known as the Herglotz-Petrovsky-Leray formula. In the case of variable coefficients, Leray corrected and extensively extended the results of Petrovsky based upon the energy method. The notes of lectures that he delivered at Princeton University in New Jersey in 1953 and Rome in 1956 have inspired much of the subsequent work.

**Analytic Partial Differential Equations** . Between 1955 and 1965, Leray initiated the study of Cauchy’s problem for analytic partial differential equations that are singular on the manifold carrying the initial data. The goal was to prove that the singularities of the solution belong to the characteristics issued from the singularities of the data or tangent to the manifold carrying them. This program, not fully realized yet, led Leray to new continuations of the Laplace transform, to original insights on asymptotic wave theory (with Lars Gaarding and Takeshi Kotake), and to an important generalization of Cauchy’s formula and of the residues theorem to analytic functions of several complex variables now reffered to as Cauchy-Fantappié-Leray formulas.

**Elasticity and Monotone Operators** . The decade from the mid-1950s to the mid-1960s was also enriched by important work in elasticity theory—motivated by new techniques in the construction of bridges—and in fixed-point theory, where Leray simplified and extended his earlier work by introducing what is now called the Leray trace. With Jacques-Louis Lions, the theory of monotone-like operators in Banach spaces was developed, leading to the important class of Leray-Lions operators, and paving the way for Haim Brezis’s fruitful concept of pseudo-monotone operator.

**Nonstrict Hyperbolic Systems** . The next five years were mainly devoted to the study of nonstrict hyperbolic systems, which are important, for example, in relativistic magnetohydrodynamics. In collaboration with Yujiro Ohya and Lucien Waelbroeck, Leray solved them in some Gevrey spaces, which are intermediate between the spaces of holomorphic and of smooth functions.

At the beginning of the 1970s, Vladimir I. Arnold called Leray’s attention to Victor P. Maslov’s work on asymptotic solutions of partial differential equations, connected to the WKB method in quantum mechanics. Leray brought in the techniques of pseudo-differential operators and a new structure based upon symplectic geometry and called Lagrangian analysis. His approach leads mathematically to a constant, which, in the special cases of Schrödinger, Klein-Gordon, and Dirac equations, can be identified to Max Planck’s one.

Schrödinger’s equation for one electron was solved using Fuchs’s theorem. In the early 1980s, Leray extended it to cover the case of several electrons, and describes the behavior of the solutions near the atomic nucleus. In the meantime, Leray had proved, together with Yusaku Hamada, Claude Wagschal, and Akira Takeuchi, several extensions of the Cauchy-Kovalevskaya theorem, including ramified data, and the analytic continuation of the solutions. Motivated by soil mechanics, Leray also contributed, around the beginning of the 1990s, to the propagation of waves in an elastic half-plane, through a new technique called the Laplace-d’Alembert transform.

## BIBLIOGRAPHY

*A comprehensive bibliography of Leray’s work can be found in each volume of Jean Leray*, Oeuvres scientifiques. *3 vols. Paris: Société Mathématique, 1998*.

### WORKS BY LERAY

*Hyperbolic Differential Equations*. Princeton, NJ: Institute for Advanced Studies, 1953.

*Notice sur les travaux scientifiques de M. Jean Leray*. Paris: Gauthier-Villars, 1953.

*La théorie de Garding des équations hyperboliques linéaires*. Rome: Istituto di Alta Matematica, University of Rome; Varenna, Italy: CIME, 1956.

*Analyse lagrangienne et mécanique quantique*. Séminaire du Collège de France (1976–1977), I, Exposé No. 1, 303 p., Paris: Collège de France. Translated by Carolyn Schroeder as *Lagrangian Analysis and Quantum Mechanics: A Mathematical Structure Related to Asymptotic Expansions and the Maslov Index*. Cambridge, MA: MIT Press, 1982.

*Oeuvres scientifiques*. 3 vols. Paris: Société Mathématique, 1998.

### OTHER SOURCES

Andler, Martin. “Jean Leray.” *Proceedings of the American Philosophical Society* 144 (2000): 469–478.

Guillopé, Laurent. *Actes des journées mathématiques à la mémoire de Jean Leray*. (Nantes 2002), Séminaires et congrès no. 9, Société mathématique de France, 2004.

“Jean Leray (1906–1998).” *Gazette des Mathématiciens* 84, suppl., (2000). This is a special issue of the *Gazette des Mathématiciens*, issued by the Société mathématique de France as a supplement to no. 84. It contains articles about Leray by Kantor, Choquet-Bruhat, Siegmund-Schultze, Miller, Houzel, Yger, Serrin, Chemin, and Malliavin.

Lions, Jacques-Louis. “Les travaux de Jean Leray en mecanique des fluids.” *Gazette des mathematiciens* 75 (1988): 7–8.

Mawhin, Jean. “In Memoriam Jean Leray (1906–1998).” *Topological Methods in Nonlinear Analysis* 12 (1998): 199–206.

———. “Jean Leray (1906–1998).” *Académie. Royale de Belgique. Bulletin de la Classe des Sciences*, series 6, 10 (1999): 89–98.

———. “Leray-Schauder Degree: A Half Century of Extensions and Applications.” *Topological Methods in Nonlinear Analysis* 14 (1999): 195–228.

Schmidt, Marian, ed. *Hommes de Science: 28 Portraits*. Paris: Herrmann, 1990.

*Jean Mawhin*

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