User:DVD206/The Laplace-Beltrami operator and harmonic functions

• a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation.
• mean-value property

The value of a harmonic function is a weighted average of its values at the neighbor vertices.

• maximum principle

Corollary: the maximum (and the minimum) of a harmonic functions occurs on the boundary of the graph or the manifold.

• harmonic conjugate

One can use the system of Cauchy Riemann equations

${\begin{cases}\gamma u_{x}=v_{y},\\\gamma u_{y}=-v_{x}\end{cases}}$ to define the harmonic conjugate

• analytic continuation

Analytic continuation is an extension of the domain of a given analytic function.