A (nonconstant) harmonic function takes its maximum value and its minimum value on the boundary of any region (not at an interior point). Thus, for example, the electrostatic potential V in a region containing no free charge takes on its largest and smallest values on the boundary of the region; similarly, the temperature T of a body containing no sources of heat takes its largest and smallest values on the surface of the body. Prove this fact (for two-dimensional regions) as follows: Suppose that it is claimed that u(x, y) takes its maximum value at some interior point a; this means that, at all points of some small disk about a, the values of u(x, y) are nolarger than at a. Show by Problem 36 that such a claim leads to a contradiction (unless u = const.). Similarly prove that u(x, y) cannot take its minimum value at an interior point.
A (nonconstant) harmonic function takes its maximum value and its minimum value on the boundary of any region (not at an interior point). Thus, for example, the electrostatic potential V in a region containing no free charge takes on its largest and smallest values on the boundary of the region; similarly, the temperature T of a body containing no sources of heat takes its largest and smallest values on the surface of the body. Prove this fact (for two-dimensional regions) as follows: Suppose that it is claimed that u(x, y) takes its maximum value at some interior point a; this means that, at all points of some small disk about a, the values of u(x, y) are no
larger than at a. Show by Problem 36 that such a claim leads to a contradiction (unless u = const.). Similarly prove that u(x, y) cannot take its minimum value at an interior point.
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