Partial differential equations

elective courses

It is recommended to take Functional analysis prior (or, at least, in parallel) to this course.

Introduction to the theory of linear partial differential equations. The first part focuses on the classical theory. The second gives introduction to the theory of Sobolev spaces and weak solutions of elliptic problems. The course does not require previous experience in PDEs. However, the students are encouraged to take a basic course in Functional Analysis, at least parallelly. Certain issues, in particular concerning the theory of weak solutions, will be treated in a more detailed way compared to the course "Introduction to PDE".

Examples of partial differential equations, physical motivations. Equations of the first order and the method of characteristics (2 lectures).

Classical theory of elliptic equations (representation of solutions, maximum principles and their applications) (3 lectures).

Fourier Transform: definition, basic properties and applications to linear PDE (1 lecture).

Introduction to the theory of distributions and Sobolev spaces: notion and basic properties of weak derivatives, trace and imbedding theorems, the Rellich-Kondrashov Theorem (3 lectures).

Lax-Milgram Lemma and its applications in existence proofs for weak solutions of elliptic problems, Galerkin Method for linear problems (2 lectures).

Introduction to spectral theory. Fredholm alternative and its applications to linear elliptic problems (1 lecture).

Information on the regularity theory of elliptic equations. Examples of applications of fixed point theorems and Galerkin method to nonlinear problems.

(1 lecture).

L.C.Evans, Partial differntial equations. AMS 1998

D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin 1983

A student:

1. Knows basic properties of the Laplace operator and harmonic functions.

2. Knows the definition and basic properties of the Fourier transform and examples of its applications to linear PDE.

3. Knows basic properties of Sobolev spaces. Is aware of basic versions of trace and imbedding theorems

and knows how to apply them in the estimates for linear PDE.

4. Is able to apply the Lax-Milgram Theorem to prove the existence of weak solutions to elliptic problems.

5. Knows the Galerkin method and its basic applications in PDE.

6. Is familiar with classical fixed point theorems and their applications to simple nonlinear equations.