User:DVD206/Harmonic functions

Harmonic functions on graphs and manifolds describe stationary distribution of random processes such as random walk and Brownian motion respectively. There are discrete, given by difference equation and continuous, given by differential equations models of definitions of harmonic functions, on graphs and manifolds respectively.

A function u on the vertices of a graph w/boundary is harmonic if its value at every interior vertex is the average of its values at neighboring vertices. That is,

$u(v)=\sum _{v\rightarrow p}\gamma (vp)u(p)/\sum _{v\rightarrow p}\gamma (vp).$

• a harmonic function on a manifold is a twice continuously differentiable function u : U → R (where U is an open subset of R²) which satisfies Laplace's equation:

$\nabla \cdot (\gamma \nabla u)=(\gamma u_{x})_{x}+(\gamma u_{y})y)=0.$

Definition of the Dirichlet-to-Neumann operator for a domain

The harmonic functions satisfy the following properties:

• 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.

Harmonic/analytic continuation on a square grid