Elliptic Boundary Value Problem - The Main Example

The Main Example

In two dimensions, let be the coordinates. We will use the notation for the first and second partial derivatives of with respect to, and a similar notation for . We will use the symbols and for the partial differential operators in and . The second partial derivatives will be denoted and . We also define the gradient, the Laplace operator and the divergence . Note from the definitions that .

The main example for boundary value problems is the Laplace operator,

where is a region in the plane and is the boundary of that region. The function is known data and the solution is what must be computed. This example has the same essential properties as all other elliptic boundary value problems.

The solution can be interpreted as the stationary or limit distribution of heat in a metal plate shaped like, if this metal plate has its boundary adjacent to ice (which is kept at zero degrees, thus the Dirichlet boundary condition.) The function represents the intensity of heat generation at each point in the plate (perhaps there is an electric heater resting on the metal plate, pumping heat into the plate at rate, which does not vary over time, but may be nonuniform in space on the metal plate.) After waiting for a long time, the temperature distribution in the metal plate will approach .