On 2D Inverse Problems/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,

$u(p)=\sum _{p\to q}\gamma (pq)u(q)/\sum _{p\to q}\gamma (pq).$

Or, alternatively, u satisfies Kirchhoff's law for potential at every interior vertex p:

$\sum _{p\to q}\gamma (pq)(u(p)-u(q))=0.$

A harmonic function on a manifold M is a twice continuously differentiable function u : M → R, where u satisfies Laplace equation:

${\mathrm{\Delta }}_{\gamma }u=\mathrm{\nabla }\cdot (\gamma \mathrm{\nabla }u)=0.$

A harmonic function defined on open subset of the plane satisfies the following differential equation:

$(\gamma u_x{)}_x+(\gamma u_y{)}_y=0.$

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

$\{\begin{array}{c}\gamma u_x=v_y,\\ \gamma u_y=-v_x\end{array}$

to define the harmonic conjugate.

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

Harmonic/analytic continuation on a square grid

Dirichlet problem

Harmonic functions minimize the energy integral or the sum

$\underset{\mathrm{\Omega }}{\int }\gamma |\mathrm{\nabla }u{|}^2\text{ and }\sum _{e=(p,q)\in E}\gamma (e)(u(p)-u(q){)}^2$

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.