Georgia Institute of Technology College of Sciences

A harmonic function of two variables is the real or imaginary part of an analytic function. A harmonic function of $n$ variables is a function $u$ satisfying

$$

\frac{\partial^2 u}{\partial x_1^2}+\ldots+\frac{\partial^2u}{\partial x_n^2}=0.

$$

We will first recall some basic results on harmonic functions: the mean value property, the maximum principle, the Liouville theorem, the Harnack inequality, the Bocher theorem, the capacity and removable singularities. We will then present a number of more recent results on some conformally invariant elliptic and degenerate elliptic equations arising from conformal geometry. These include results on Liouville theorems, Harnack inequalities, and Bocher theorems.