Partial Differential Equations in the 20th Century

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 18901900: 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. LeraySchauder 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 CalderonZygmund theory. 16. Estimates for general linear elliptic boundary value problems. 17. Linear equations of evolution: The HilleYosida theory. 18. Spectral theories. 19. Maximum principle and applications: The DeGiorgiNash 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. * A version of this article will appear in italian translation in the Enciclopedia Italiana in its series on the history of the 20th century.