Potential Theory

Schedule

Monday 5:10-7 pm
Wednesday 6:10-7 pm

Location: Virtual Reality (should be working this time!)

First class: Oct. 19, 2020

About the course

Potential theory lies in the intersection of complex analysis, probability and partial differential equations. In this course, we study potential theory from all three perspectives and show how these fields interact. One of the fundamental notions in this course is that of polar set. Polar sets are negligible from the point of view of potential theory, just like measure 0 sets are negligible in real analysis.

PROBLEMS

The second part of the course will focus on non-linear potential theory.

Slides: Basic properties of harmonic functions (updated)

Slides: A Primer on Brownian Motion

Slides: Harmonic Measure and Green's functions

Slides: Some martingale arguments

Slides: Subharmonic functions

Slides: Perron's Method

Slides: Potentials and Energy

Slides: Frostman's theorem and Applications

Slides: Sobolev spaces and Dirichlet's principle

Notes: Liouville's theorem

Notes: Vasquez theorem

List of topics

1 What is potential theory? Basic properties of harmonic functions. Using Brownian Motion to solve the Dirichlet problem. Recurrence and transience. Convergence of bounded martingales. Further applications to complex analysis.
2 The Poisson kernel. No positive harmonic functions in R^n. Harmonic functions are real analytic. Harnack's principle.
3 Basic properties of subharmonic functions. Phragmen-Lindelof principle. Liouville's theorem for subharmonic functions.
4 Distributions. Weyl's lemma. Riesz decomposition. Potentials. Energy. Equilibrium measures. Polar Sets. Frostman's theorem. Brelot-Cartan theorem. Minus infinity sets. Removable singularities.
5 Capacity and transfinite diameter. Hilbert lemniscate theorem.
6 Schauder's fixed point theorem. Existence and uniqueness of solutions of the Gauss curvature equation with prescribed boundary data. Liouville's theorem on the correspondence between conformal metrics of curvature -4 and holomorphic self-maps of the unit disk. Heins theorem on prescribing finite Blaschke products by their critical points.
7 Basic properties of Sobolev spaces. Regularity of solutions of the linear Dirichlet problem with singularities.
8 Maximal functions. Sobolev capacities. Removable sets for Sobolev functions. Obstacle problems. Carleson's theorem on removable sets of Holder continuous harmonic functions.

Main references

Additional references