Louis Nirenberg

Louis Nirenberg in Jerusalem , 1975

Louis Nirenberg (born February 28, 1925 in Hamilton , Ontario ; † January 26, 2020 in New York City ) was a Canadian mathematician who mainly researched the field of partial differential equations .

Life

Nirenberg attended high school in Montreal and studied at the McGill University (bachelor's degree in 1945), then at the New York University , where he at Richard Courant and Kurt Friedrichs studied, in 1947 received his master's degree and in 1949 James Stoker Ph.D. (in it he solved Weyl's embedding problem of differential geometry). 1951/52 he was in Zurich with Heinz Hopf and in Göttingen with Franz Rellich among others . Nirenberg became a professor at the Courant Institute of Mathematical Sciences , where he stayed for the rest of his career until his retirement in 1999 and was temporarily its director. Nirenberg supervised over 40 doctoral students there. In 1958 he was at the Institute for Advanced Study .

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Nirenberg is considered one of the outstanding analysts of the 20th century. He made fundamental contributions to the theory of linear and nonlinear partial differential equations and their applications in differential geometry and complex analysis.

With Fritz John , he began the study of functions with "bounded mean oscillation" (BMO) : ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\frac {1}{\left|V_{Q}\right|}}\int _{Q}\left|u-{\hat {u}}_{Q}\right|{\rm {d}}V\leq K}$,

where integration and mean values ​​are considered in cubes with volume and is a constant. They showed that these are of "exponential class" (with a constant ): ${\displaystyle {\hat {u}}_{Q}}$ ${\displaystyle Q}$ ${\displaystyle V_{Q}}$ ${\displaystyle K}$ ${\displaystyle \lambda }$

${\displaystyle \int \exp {(\lambda \left|u-{\hat {u}}_{Q}\right|)}{\rm {d}}V<\infty }$.

With Luis Caffarelli and Robert V. Kohn , he investigated the possible singularities in the Navier-Stokes equations (a problem that is still largely open and has been included in the list of Millennium Problems of the Clay Mathematics Institute ). They characterized it by the rate of concentration of the energy density around the possible singular points and showed that the 1-dimensional Hausdorff measure of the singular points disappears in three spatial dimensions. In doing so, they built on the work of Vladimir Scheffer from the mid-1970s.

With his doctoral student August Newlander, he characterized complex structures among almost complex structures in the (Newlander-Nirenberg Theorem). They showed that integrability conditions, which generalize the Cauchy-Riemann equations in the case , are not only necessary but also sufficient. As Nirenberg remembers, he was inspired to this problem by André Weil and Shiing-Shen Chern - in particular Weil challenged the analysts who deal with partial differential equations to also, once in his view, really fundamental problems, in this case a long open problem of complex analysis. With the theorem of Newlander-Nirenberg, he and Kodaira and Donald Spencer proved existential theorems about the deformation of complex structures. ${\displaystyle \mathbb {R} ^{2n}}$ ${\displaystyle n=1}$

In a 1965 work with Joseph Kohn , he introduced pseudo differential operators . According to her own statements, this was a by-product of her work on the - Neumann problem , which required previously unpublished results on the algebra of singular integral operators. ${\displaystyle {\overline {\partial }}}$

Characteristic of the work of Nirenberg is (as with his teacher Kurt Friedrichs, described the Nirenberg in an interview in 2002 as the mathematician who him most affected) are often an artful use of inequalities, for example, in working with Avron Douglis and Shmuel Agmon over Estimates of boundary value problems of elliptical partial differential equations, based on the work of Juliusz Schauder . He did not see himself as the founder of theoretical buildings, but as a problem solver.

Honors and memberships

He has received numerous honors and prizes, first in 1959 with the AMS Bôcher Memorial Prize for “outstanding achievements in mathematical analysis” . In 1962 he gave a plenary lecture at the International Congress of Mathematicians in Stockholm (Some Aspects of linear and nonlinear partial differential equations). He was a Guggenheim Fellow (1966) and Sloan Fellow and received the “Award of Excellence in Science and Technology” from the City of New York. In 1982 he was the first to receive the Swedish Crafoord Prize (together with Vladimir I. Arnold ) , the Leroy P. Steele Prize of the American Mathematical Society in 1994 , the National Medal of Science in 1995 and the first Chern Medal of the IMU in 2010 . In 2014 he was awarded the Leroy P. Steele Prize for his work with Robert V. Kohn and Caffarelli in 1982. In 2015 he received the Abel Prize, one of the most important mathematics prizes ever.

Nirenberg was a member of the National Academy of Sciences of the USA, the American Academy of Arts and Sciences (1965), the American Philosophical Society , the Norwegian Academy of Sciences , the Ukrainian and Lombard Academy of Sciences, the Paris Académie des Sciences , the Italian Accademia dei Lincei and the Royal Society of Canada (2011). He was a fellow of the American Mathematical Society . The McGill University (1986), the University of Pisa (1990), the University of Paris-Dauphine (1990), the McMaster University (2000) and the University of British Columbia (2010) awarded him an honorary doctorate.

Fonts

• Lectures on linear partial differential equations. In: Conference Board of the Mathematical Sciences of the AMS. American Mathematical Society, Providence (Rhode Island) 1973.
• Functional Analysis. Courant Institute 1961.
• Topics in Nonlinear Functional Analysis. Courant Institute 1974.
• Partial differential equations in the first half of the century , in Jean-Paul Pier Development of mathematics 1900-1950 , Birkhäuser 1994

Web links

• Louis Nirenberg in the Mathematics Genealogy Project (English)
• Interview with Louis Nirenberg in Notices of the AMS 2002 (PDF; 655 kB)
• National Medal of Science for Nirenberg, Notices AMS - with recognition of his work by Luis Caffarelli and Joseph Kohn (PDF; 180 kB)
• Nirenberg on his work with Caffarelli and Kohn on the Navier-Stokes equations, 1981
• Chern Medal for Nirenberg 2010
• Tristan Riviere: Exploring the Unknown: The Work of Louis Nirenberg on Partial Differential Equations, Notices AMS, February 2016 (and interview with Nirenberg)
• Videos by and about Louis Nirenberg in the AV portal of the Technical Information Library
• Nature: Louis Nirenberg (1925-2020): Virtuoso mathematician who transformed the study of partial differential equations . OBITUARY February 17, 2020

Individual evidence

1. ^ Death of Louis Nirenberg
2. ^ ^{A b} NYU Courant Mourns the Loss of Professor Louis Nirenberg. NYU Courant Institute of Mathematical Sciences, accessed January 31, 2020 . 
3. ^ Louis Nirenberg: The Weyl and Minkowski Problems in Differential Geometry in the Large. In: Comm. Pure Applied Math. Volume 6, 1953, pages 337-394.
4. ↑ Allyn Jackson, Notices AMS, April 2002
5. ^ Fritz John, Louis Nirenberg: On functions of bounded mean oscillation. In: Comm. Pure Applied Math. Volume 14, 1961, pages 415-426.
6. ↑ singular sets for weak solutions of the Navier-Stokes equations; the consideration of weak solutions first followed Jean Leray , who proved their existence in three dimensions
7. ^ Luis Caffarelli, Robert V. Kohn, Louis Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes Equations. In: Comm. Pure Applied Math. Volume 35, 1982, pages 771-831.
8. ↑ See the representation of Fefferman Millennium Problem Navier-Stokes equations, PDF file ( Memento June 13, 2010 in the Internet Archive ). The proof of the theorem of Caffarelli, Kohn, Nirenberg is from F.-H. Lin ( A new proof of the Caffarelli-Kohn-Nirenberg theorem. In: Comm. Pure and Applied Mathematics. Volume 51, 1998, pages 241-257).
9. ^ A. Newlander, Louis Nirenberg: Complex analytic coordinates in almost complex manifolds. In: Annals of Mathematics. Volume 65, 1957, pages 391-404.
10. ↑ Kunihiko Kodaira, Donald Spencer, Louis Nirenberg: On the existence of deformations of complex analytic structures. In: Annals of Mathematics. Volume 68, 1958, pages 450-459.
11. ^ Kohn, Nirenberg, An algebra of pseudodifferential operators , J. Pure Applied Mathematics, Volume 18, 1965, pp. 269-305
12. ↑ Interview Notices AMS 2002
13. ↑ Interview, Notices AMS 2002, No. 4, p. 442. His view of mathematics very much formed my view ... He was a great lover of inequalities, and that affected me very much.
14. ↑ Agmon, Douglis, Nirenberg "Estimates near the boundary of solutions of elliptic partial differential equations under general boundary conditions", Comm. Pure Applied Math., Vol. 12, 1959, pp. 623-727
15. ↑ Interview Notices AMS 2002, loc.cit.
16. ↑ Caffarelli, Kohn, Nirenberg Partial regularity of suitable weak solutions of the Navier-Stokes equations , Communications Pure and Applied Mathematics, Volume 35, 1982, pp. 771-831