Bounded harmonic functions

Bounded harmonic functions are mathematical functions that describe smooth, steady-state situations—like temperature distribution or potential fields—that do not fluctuate wildly and stay within certain limits. These functions satisfy a special property called "harmonicity," meaning their value at any point is equal to the average of their values nearby. Being "bounded" means their values stay within a fixed range, preventing extreme highs or lows. In essence, bounded harmonic functions model stable, balanced systems where the state is smoothly distributed and remains within predictable bounds, often used in physics and engineering to analyze heat flow, electrostatics, and more.