The PDE.

  We will assume that the conductive properties of the plate are independent of direction, location, and temperature. Then the PDE governing the temperature distribution T(x,y) in the interior of the plate is the Laplace equation,  

$$\nabla^2 T \equiv \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0.$$ (15)

It is the standard example of an elliptic equation. The Laplace equation also governs incompressible potential flows, electrostatics, shapes of stretched membranes, minimal surfaces, differentiable complex functions, and many other areas.