### Math 554: Linear Analysis and Partial Differential Equations

• Instructor: Richard Laugesen
• e-mail: laugesen@illinois.edu
• Office: 376 Altgeld Hall Phone: 333-1329
• Office hours: Monday 4-5, Tuesday 4-5, Thursday 3-4 or by email appointment.
• Class: MWF 2-3pm, in 343 Altgeld Hall
• Text: Partial Differential Equations by L.C. Evans. (Errata: `http://math.berkeley.edu/~evans/`). Eigenfunction of the Laplacian on an L-shaped region. (The Mathworks.)

### Course outline

We develop a modern framework for linear partial differential equations. Then we apply it to second-order elliptic, parabolic, and (briefly) hyperbolic equations, such as the kind of potential, diffusion and wave equations that arise in nonhomogeneous media. The course will be valuable to students of differential equations, analysis, probability, and differential geometry.

We cover the following topics:

• Sobolev spaces: these provide the context in which most modern research on partial differential equations is conducted.
• Second-order elliptic equations: Lax-Milgram theorem, existence of weak solutions, regularity, maximum principle, spectral theory of self-adjoint operators (including properties of low eigenvalues, and high-energy asymptotics).
• Second-order parabolic equations: existence of weak solutions, regularity, maximum principle.
• Second-order hyperbolic equations: existence of solutions; propagation of waves.
• Linear semigroup theory for evolution equations.
Math 554 obtains qualitative information on partial differential equations even when explicit solution formulas do not exist.

### Prerequisites

A strong grasp of basic real analysis (Math 447 or equivalent), and fluency with multivariable calculus are essential. Some measure theory (Math 540) would be very helpful, as would familiarity with partial differential equations (such as Math 442, or preferably Math 553).