Autumn 2007

MAT-INF4300: Partial differential equations and Sobolev spaces I

MAT-INF4300: Partial differential equations and Sobolev spaces I

Main book:

L. C. Evans. Partial differential equations.

Alternative reading:

G. B. Folland. Introduction to partial differential equations.

J. Jost. Partial differential equations.

M. E. Taylor. Partial differential equations. Basic theory.

Lecture 1 (20.08)

The Laplace equation. Properties of harmonic functions.

Sections: 2.2.2-2.2.3a.

Lecture 2 (23.08)

Properties of harmonic functions.

Sections: 2.2.3b-2.2.3d.

Exercises: 1.5.2-1.5.4 and these three.

Lecture 3 (27.08)

Green's functions.

Sections: 2.2.1a, 2.2.4a.

Lecture 4 (30.08)

Green's functions for the half-space and the unit ball.

Sections: 2.2.4b, 2.2.4c.

Exercises: 2, 4, 6, 9 in section 2.5 and this one.

Lecture 5 (13.09)

An overview of measure theory.

Lecture 6 (17.09)

Weak derivatives. Sobolev spaces.

Sections: 5.2.1, 5.2.2.

Exercises: these two.

Lecture 7 (21.09)

Approximation by smooth functions.

Sections: 5.2.3-5.3.3.

Lecture 8 (24.09)

Extensions.

Sections: 5.4.

Lecture 9 (27.09)

Extensions. Traces.

Sections: 5.4, 5.5.

Exercises: 7, 14, 16 in section 5.10 and this one.

Lecture 10 (1.10)

Sobolev inequalities.

Sections: 5.1, 5.6.1.

Lecture 11 (4.10)

Sobolev inequalities.

Sections: 5.6.1, 5.6.2.

Exercises: 1, 3, 10 in section 5.10 and this one.

Lecture 12 (8.10)

Compact operators. The Arzela-Ascoli theorem.

Compulsory assignment.

Lecture 13 (11.10)

The Rellich theorem.

Sections: 5.7.

Exercises: 9, 15, 17 in section 5.10.

Lecture 14 (15.10)

Poincare's inequality. Differentiability almost everywhere.

Sections: 5.8.1, 5.8.3.

Lecture 15 (22.10)

Weak solutions. Hilbert space.

Sections: 6.1.2, 6.2.1.

Lecture 16 (25.10)

The Lax-Milgram theorem. Elliptic equations.

Sections: 6.1.1, 6.2.1, 6.2.2.

Exercises: these two.

Lecture 17 (29.10)

Interior regularity.

Sections: 6.3.1.

Lecture 18 (1.11)

Interior regularity. Boundary regularity.

Sections: 6.3.1, 6.3.2.

Exercises: 1, 3 in section 6.6.

Lecture 19 (5.11)

Boundary regularity.

Sections: 6.3.2.

Lecture 20 (12.11)

Spectrum of elliptic operators.

Sections: 6.2.3.

Lecture 21 (15.11)

Formally self-adjoint elliptic operators.

Sections: 6.5.1.

Lecture 22 (19.11)

Maximum principle.

Sections: 6.4.

Exercises: these three.

Lecture 23 (22.11)

Eigenvalues of formally self-adjoint elliptic operators.

Sections: 6.5.1.

Lecture 24 (26.11)

Eigenvalues of the Laplacian: an overview.

Solutions to the exam.