Winter 2026, Math 251B: Topics in PDEs

 MWF 11am at MS 6201

Office hours: TBA at MS 7923

We will focus on Elliptic PDEs, both in linear and nonlinear setting, often motivated by variational problems.

 Topics to be discussed:

   •   Linear elliptic PDEs, regularity.

   • Nonlinear Elliptic problem: Hilbert’s XIXth problem, De Giorgi theory

• Energy minimizers: Euler-Lagrange equation, existence, convex duality, constraints.
• (If time permits) BV space and geometric variational problems.
• Evolution PDEs and gradient flows.

   We will assume that students are familiar with basic measure theory, including the Sobolev spaces. We will discuss any other necessary preliminary material as we proceed.

Homework: There will be 3-4 sets of homework problems.

References:

Fernandez-Real and Ros-Oton,  Regularity theory for Elliptic PDE.  https://arxiv.org/abs/2301.01564

Struwe, Variational Methods.

Evans, Partial Differential Equations.

Santambrogio, Gradient flows: an Overview, Bulletin of Mathematical Sciences 7, 87- 104 (2017).