Monday Math 49

The Laplacian operator ∇2 appears in a number of important partial differential equations. The simplest among these is Laplace’s equation: $\nabla^2f=0$. Solutions to this equation are known as harmonic functions. The linear combination of two or more harmonic functions is also harmonic (superposition principle). One generally solves Laplace’s equation over a region given the values of the function on the boundary (Dirichlet problem), or the values of the normal derivative of the function on the boundary (Neumann boundary condition).

An important property of harmonic functions is the maximum principle: for a harmonic function on some region, the maximum value occurs on the boundary, and if it occurs anywhere in the interior, the function is constant. A similar principle applies to minima. In general, this arises from the fact that one cannot have local extrema in a harmonic function; consider the second partial derivative test, and the trace of the Hessian.

In two-dimensions, with Cartesian coordinates, Laplace’s equation expands to $\frac{\partial^2f}{\partial{x}^2}+\frac{\partial^2f}{\partial{y}^2}=0$. Let us define the complex number z=x+iy, and consider an analytic function of z: $f(z)=\phi(x,y)+\imath\psi(x,y)$. Now, the consition for the function to be analytic is that φ and ψ obey the Cauchy-Riemann equations:
$\frac{\partial\phi}{\partial{x}}=\frac{\partial\psi}{\partial{y}}$
$\frac{\partial\phi}{\partial{y}}=\frac{\partial\psi}{\partial{x}}$
(see here).
Taking the partial derivative of the first equation with respect to x.
$\begin{eqnarray}\frac{\partial}{\partial{x}}\frac{\partial\phi}{\partial{x}}&=&\frac{\partial}{\partial{x}}\frac{\partial\psi}{\partial{y}}\\\frac{\partial^2\phi}{\partial{x}^2}&=&\frac{\partial^2\psi}{\partial{x}\partial{y}}\\\frac{\partial^2\phi}{\partial{x}^2}&=&\frac{\partial^2\psi}{\partial{y}\partial{x}}\\\frac{\partial^2\phi}{\partial{x}^2}&=&\frac{\partial}{\partial{y}}\frac{\partial\psi}{\partial{x}}\\\frac{\partial^2\phi}{\partial{x}^2}&=&\frac{\partial}{\partial{y}}\left(-\frac{\partial\phi}{\partial{y}}\right)\\\frac{\partial^2\phi}{\partial{x}^2}&=&-\frac{\partial^2\phi}{\partial{y}^2}\\\frac{\partial^2\phi}{\partial{x}^2}+\frac{\partial^2\phi}{\partial{y}^2}&=&0\end{eqnarray}$
and similarly,
$\begin{eqnarray}\frac{\partial}{\partial{y}}\frac{\partial\phi}{\partial{x}}&=&\frac{\partial}{\partial{y}}\frac{\partial\psi}{\partial{y}}\\\frac{\partial^2\phi}{\partial{y}\partial{x}}&=&\frac{\partial^2\psi}{\partial{y}^2}\\\frac{\partial^2\phi}{\partial{x}\partial{y}}&=&\frac{\partial^2\psi}{\partial{y}^2}\\-\frac{\partial^2\psi}{\partial{x}^2}&=&\frac{\partial^2\psi}{\partial{y}^2}\\\frac{\partial^2\psi}{\partial{x}^2}+\frac{\partial^2\psi}{\partial{y}^2}&=&0\end{eqnarray}$.
Thus, both the real and imaginary components of an analytic complex function are (two dimensional) harmonic functions.
If we have a harmonic function over some space, and transform that space via a conformal map, the resulting function is also harmonic. This is very useful in two dimensions, as any analytic complex function provides a conformal map in two dimensions (see here).

Lastly, we note that in any number of dimensions, harmonic functions remain harmonic under a rotation of the coordinates.