Entry requirements

120 credits including 90 credits in mathematics with Real Analysis.

Learning outcomes

On completion of the course, the student should be able to

• account for central properties of solutions to Laplace's equation, the heat equation and wave equations;
• solve non-linear equations of the first order with the method with characteristics;
• account for Sobolev spaces and central properties of Sobolev functions like theorems concerning approximation, extension and traces as well as the Sobolev inequalities and theorems concerning compactness;
• account for existence - and uniqueness theorems of weak solutions to elliptic equations of the second order;
• account for regularity theory, maximum principles and eigenvalues/eigenfunctions for the elliptic equations of the order;

Content

Laplace equation. Heat equation. The wave equation. Non-linear equations of the first order. The methods with characteristics. Some methods to design explicit solutions. Introduction to Functional Analysis relevant to the course. Sobolev spaces and approximation theorems for Sobolev functions. Extension and trace theorems. The Sobolev inequalities and theorems concerning compactness. Existence and uniqueness of weak solutions to elliptic equations of second order. Regularity theory, maximum principles and eigenvalues/eigenfunctions for the elliptic equations of second order.

Instruction

Lectures,problem solving sessions, student presentations.

Assessment

Assignments and oral examination.

Other directives

The course can not be included in higher education qualification together with Partial Differential Equations, advanced course, or the equivalent.