User:DVD206/Printed Version/Harmonic functions
Harmonic functions can be defined as solutions of differential and difference Laplace equation as follows.
A function/vector u defined on the vertices of a graph w/boundary is harmonic if its value at every interior vertex p is the average of its values at neighboring vertices. That is,
Or, alternatively, u satisfies Kirchhoff law for potential at every interior vertex p:
Harmonic function on a manifold M is a twice continuously differentiable function u : M → R, where u satisfies Laplace equation:
Harmonic function defined on open subset of the plane satisfies the following differential equation:
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,
the system of Cauchy-Riemann equations,
can be used to define the harmonic conjugate,
is an extension of the domain of definition of a given harmonic function.
Dirichlet problem
[edit | edit source]Harmonic functions minimize the energy integral or the sum:
if the values of the functions are fixed at the boundary of the domain or the network in the continuous and discrete models respectively. The minimizing function/vector is the solution of the Dirichlet problem with the prescribed boundary data.