On 2D Inverse Problems/Harmonic functions

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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's law for potential at every interior vertex p:

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

A 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

One can use the system of Cauchy-Riemann equations

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

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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.