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Partial differential equations in the 20th century. (English) Zbl 0915.01011

The article presents a broad survey of various aspects of the present day theory of PDE’s preceded by a brief historical introduction starting with the 18th and 19th centuries; a version of the article is to appear in Italian translation in the Enciclopedia Italiana “in its series on the history or the 20th century”. A good idea of its scope is given by its table of contents: 1. Introduction 2. Models of PDE’s in the 18th and 19th century. 3. Methods of calculating solutions in the 19th century. 4. Developments of rigorous theories of solvability in the last decades of the 19th century. 5. The period 1890-1900: the beginning of modern PDE and the work of Poincaré. 6. The Hilbert programs, 7. S. Bernstein and the beginning of a priori estimates. 8. Solvability of second order linear elliptic equations. 9. Leray-Schauder theory. 10. Hadamard and the classification of PDE’s and their boundary value problems. 11. Weak solutions. 12. Sobolev spaces. 13. The Schwartz theory of distributions. 14. Hilbert space methods. 15. Singular integrals in \(L^p\); the Calderon-Zygmund theory. 16. Estimates for general linear elliptic boundary value problems. 17. Linear equations of evolution: The Hille-Yosida theory. 18. Spectral theories. 19. Maximum principle and applications: The DeGiorgi-Nash estimates. 20. Nonlinear equations of evolution: Fluid flows and gas dynamics. 21. Nonlinear PDE’s and nonlinear functional analysis. 22. Free boundary value problems: Variational inequalities. 23. Quasilinear and fully nonlinear elliptic equations. 24. PDE’s and differential geometry. 25. Computation of solutions of PDE’s: Numerical analysis and computational science.

MSC:

01A60 History of mathematics in the 20th century
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
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