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.