Bertinetto's theorem

Bertinetto's theorem concerns harmonic functions—these are functions that describe steady-state solutions, like temperature distribution, with no internal sources or sinks. The theorem states that if such a harmonic function is bounded near the boundary of a domain (a region in space), then its maximum and minimum values inside the domain are achieved on that boundary. Essentially, the theorem confirms that, under these conditions, a harmonic function's extremal values are determined by the boundary, ensuring predictable behavior within the region. This principle is fundamental in potential theory and has applications in physics, engineering, and mathematics for analyzing equilibrium states.