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:

The value of a harmonic function is a weighted average of its values at the neighbor vertices,
 * mean-value property

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

One can use the system of Cauchy-Riemann equations
 * harmonic conjugate



\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.



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.