Harmonic Functions and Beyond

A harmonic function of one variable is a linear function. 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

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