OLEINIK, OLGA ARSENIEVNA

(b.Kiev, Ukraine, USSR, 2 July 1925; d. Moscow, Russia, 13 October 2001)

mathematics, differential equations.

Oleinik was one of the few outstanding women mathematicians of the twentieth century. The general theory of partial differential equations, which describe the behavior of fluids, gases, elasticity, electromagnetism, and quantum physics, was developed during the last century, and Olga Oleinik was one of the major figures in that process. Her work began principally with elliptic equations (such as the wave equation u^xx + u^yy = 0), but her major contributions were to singular elliptic and parabolic equations and to nonlinear hyperbolic equations (such as u^tt - u^xx = 0).

Life Oleinik was born in Kiev in Ukraine. Her parents, Arseniev Ivanovic and Anna Petrovna Oleinik, lived in Matusov, a small town near Kiev, where her father was employed in a factory as a bookkeeper. Her life was uneventful until 1941, when Germany invaded the USSR in a lightning campaign. The factory was quickly moved by rail to Perm, Russia, just west of the Urals, and Olga’s father was moved with it. The family decided that Olga should accompany him and that the rest of the family would remain in Ukraine (Olga’s sister had a small infant). Olga never returned to live in Ukraine.

After completing high school in 1942, Olga attended the local university. Some of the mathematical faculty from the University of Moscow had been evacuated to Perm because of the war, and when they returned to Moscow in 1943 they arranged for Oleinik to become a student at Moscow University. There she studied with and became a lifelong friend of the mathematician Ivan Petrovsky. She attended I. Gelfand’s seminar with Olga Ladyzhenskaya and Mark Visik.

In 1947 she obtained her undergraduate degree and in 1954 her doctorate, both under the direction of Petrovsky. Her first thesis was on the topology of real algebraic curves on an algebraic surface; the second thesis was on partial differential equations, a field she stayed with, almost exclusively, for the rest of her life.

She remained at Moscow State University and on Petrovsky’s death she succeeded him to become the chair of differential equations, a post she held throughout her life. She was an early visitor to the United States, first in the early 1960s as a delegate to a women’s congress. As soon as the Soviet regime softened its restrictions, she worked indefatigably to make Western mathematical literature available to Soviet mathematicians.

Oleinik was married to Lev Alekseevich Chudov during the early 1950s and had a son. Unfortunately the son was mentally disabled, and Oleinik’s concern for him and efforts on his behalf took a considerable toll on her professional and emotional life. Oleinik died in Moscow on 13 October 2001, after a long struggle against cancer.

Mathematical Work In early work, Oleinik considered several types of elliptic equations that had singular coefficients on the boundary, which made them lose their ellipticity. A simple example of such an equation is (l-r ²)^{x uxx} + u _yy = 0 where r² =x² +y² and a>0. Her results, inspired by the work of the Italian mathematician Gaetano Fichera, are still the authority on the subject.

In her 1957 paper “Discontinuous Solutions of Non-Linear Differential Equations,” Oleinik introduced a new entropy condition, which is important for deciding which type of shocks could take place in filtration processes. It has become fundamental in the understanding of secondary oil recovery.

In the 1980s she found a greatly simplified proof of Korn’s inequality, which is fundamental in studying the equations of elasticity, and applied this result to gain new insight into the so-called St. Venant’s principle. Oleinik was one of the founders of homogenization theory for partial differential equations, which shows how to find “average” equations for a highly oscillatory system.

In a series of papers culminating in the text “Matmaticheskie metody v teoil pogranichnogo sloya” (Mathematical methods in boundary-layer theory), Oleinik proved an existence theorem for Ludwig Prandtl’s boundary layer equations. This was the basic theory in aero- and hydrodynamics, and the equations were widely used but lacked a strong mathematical framework. There had even been some question about whether there were any valid solutions. Her work also illuminated how the separation of the boundary layer of the fluid from the rigid boundary takes place.

Legacy and Honors Oleinik published more than three hundred articles and wrote eight books, one of them while recuperating from major knee surgery. She made many mathematical tours in Europe and in the United States. In 1996 she gave the Noether Lecture of the Association for Women in Mathematics at the annual meeting of the American Mathematical Society. She made extensive visits to the University of Rome and to the University of Heidelberg among many other places.

Among the many honors Oleinik received were the Lomonosov Prize in 1964, the medal of the Collège de France, and the first degree medal of Charles University in Prague. In 1995 she was awarded the Order of Honor of the Russian Federation. She was a member of the Soviet Academy of Sciences, a foreign member of the Accademia Nazionale dei Lincei in Rome, and was made doctor honoris causa of the University of Rome in 1984.

Oleinik had a great many students. Among them are N. D. Vvedenskaya, T. D. Wentzel, J. V. Egorov, G. A. Yosifian, S. Kamin, S. N. Kruzhkov, V. Petkov, E. V. Radkevich, G. A. Chechkin, Zhou Yulin, and T. A. Shaposhnikova. Most of them pursued academic and research careers.

BIBLIOGRAPHY

WORKS BY OLEINIK

“On Equations of Elliptic Type Degenerating on the Boundary of a Region” [in Russian]. Doklady Akademii Nauk SSSR, n.s., 87 (1952): 885–888.

g“Discontinuous Solutions of Non-Linear Differential Equations” [in Russian]. Uspehi Mat. Nauk, n.s., 12, no. 3 (1957): 3–73.

With V. A. Kondratiev. “Hardy’s and Korn’s Type Inequalities and Their Applications.” Rendiconti di Matematica e delle sue Applicazioni, ser. 7, 10, no. 3 (1990): 641–666.

With V. V. Jikov and S. M. Kozlov. Homogenization of Differential Operators and Integral Functionals. Translated by G. A. Yosifian. Berlin: Springer-Verlag, 1994.

With V. N. Samokhin. “Matmaticheskie metody v teoil pogranichnogo sloya” [Mathematical methods in boundary-layer theory]. Moscow: Fizmaslit “Nauka,” 1997.

OTHER WORKS

Arnold, V. I., M. I. Vishik, A. S. Kalashnikov, V. P. Maslov, S. M. Nikolskii, and S. P. Novikov. “Olga Arsenievna Oleinik (on her 70th birthday).” Trudy Seminare imeni I. G. Petrovskogo, no. 19. Translated in Journal of Mathematical Sciences85, no. 6 (1997): 2249–2259. A short biography of Oleinik and a complete bibliography of her writings from 1986 to 1997.

Jäger, Willi, Peter Lax, and Cathleen Synge Morawetz. “Olga Arsen’evna Oleinik.” Notices of the American Mathematical Society50, no. 2 (2002): 220–223.

“Olga Arsenievna Oleinik (On the Occasion of Her Sixtieth Birthday)” [in Russian]. Trudy Seminare imeni I. G. Petrovskogo 12 (1987): 3–21. For earlier publications.

Vestnik Moskovskogo Universiteta, Matematika, Mekhanika [Moscow University mathematics and mechanics bulletin] 4 (1975): 122–124 and (1985): 98–102. For earlier publications.

Cathleen Synge Morawetz