Math 619 — Fall 1999
Partial Differential Equations

Instructor: Yulij Ilyashenko

Time: TR 1:25-2:40

Room: Malott 206

Introduction. Why the general theory of PDE does not exist?

First order equations. Linear and quasilinear equations. Characteristics. Geometric theory of first order nonlinear partial differential equations (by Arnold). Relation with contact and symplectic structures.

Elements of distribution theory. Space of generalized functions. Their derivatives. Delta sequences. Fundamental solutions of linear ordinary and partial differential equations with constant coefficients. Nonhomogeneous linear equations.

Laplace equation. Fundamental solution. Properties of harmonic functions. Green function. Representation formulas for solutions of Laplace and Poisson equations.

Heat equation. Fundamental solution. Properties of solutions of the heat equation. Representation formulas. Separation of variables.

Wave equation. String: reflection and resonanes. Fundamental solutions. Representation formulas. Wave propagation. Separation of variables.

Sobolev spaces. Definition and examples. Some embedding theorems. Relations to the theory of generalized functions.

Time permitting, two and three dimensional Navier-Stokes equation will be considered.

I will try to replace computational arguments by qualitative ones. About 4/5 of the course will be covered by the books of Evans, "Partial Differential Equations," Arnold "Geometric Methods in the Theory of Ordinary Differential equations" and Kanwal "Generalized functions. Theory and technique."

Last modified: April 7, 2003