The solvability of differential equations. (English) Zbl 1232.35007
Bhatia, Rajendra (ed.) et al., Proceedings of the international congress of mathematicians (ICM 2010), Hyderabad, India, August 19–27, 2010. Vol. III: Invited lectures. Hackensack, NJ: World Scientific; New Delhi: Hindustan Book Agency (ISBN 978-981-4324-33-5/hbk; 978-81-85931-08-3/hbk; 978-981-4324-30-4/set; 978-981-4324-35-9/ebook). 1958-1984 (2011).
In [Ann. Math. 163, No. 2, 405–444 (2006; Zbl 1104.35080)] the author proved the sufficiency of the Nirenberg-Trèves condition \((\Psi)\) for the local solvability of principal type pseudo-differential equations, resolving a well-known conjecture of L. Nirenberg and F. Trèves [Commun. Pure Appl. Math. 23, 459–510 (1970; Zbl 0208.35902)]. The solvability condition \((\Psi)\) is a condition on the principal symbol only; it rules out certain sign changes. The history of the subject starts with the famous example by H. Lewy [Ann. Math. (2) 66, 155–158 (1957); erratum ibid. 68, 202 (1958; Zbl 0078.08104)] of a non-solvable first order linear differential equation. The paper under review contains an overview of the history of the solvability problem and its solution, a sketch of the proof of the sufficiency of \((\Psi)\), and an outlook with open problems.
For the entire collection see [Zbl 1220.00033].
For the entire collection see [Zbl 1220.00033].
Reviewer: Sönke Hansen (Paderborn)
MSC:
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 47G30 | Pseudodifferential operators |
| 58J40 | Pseudodifferential and Fourier integral operators on manifolds |
| 35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |
