# 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,

$u(p) = \sum_{p\rightarrow q} \gamma(pq)u(q)/\sum_{p\rightarrow q} \gamma(pq).$

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

$\sum_{p\rightarrow q} \gamma(pq)(u(p) - u (q)) = 0.$

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

$\Delta_\gamma u = \nabla\cdot(\gamma\nabla u) = 0.$

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

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

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

$\int_{\Omega}\gamma|\nabla u|^2 \mbox{ and } \sum_{e=(p,q)\in E} \gamma(e)(u(p) - u (q))^2$

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.