Partial Differential Equations in the 20th Century

✍ Scribed by Haı̈m Brezis; Felix Browder

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 605 KB
• Volume: 135
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1713

✦ Synopsis

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 Poincare . 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.