Gauss's Mean Value Property states that for a harmonic function u on a domain D, the value at any interior point z₀ equals the average of u over any circle centered at z₀ and contained in D, proved using Green's theorem and the defining PDE of harmonicity (Laplace's equation). This leads to the Strict Maximum Principle — a bounded harmonic function that attains its maximum modulus at an interior point of a simply connected domain must be constant — and its corollary, the Maximum Modulus Principle, that a harmonic function continuous on a closed bounded domain attains its maximum modulus on the boundary, not the interior, in contrast to real-variable extrema which can occur at interior critical points.
Gauss's Mean Value Property states that for a harmonic function u on a domain D, the value at any interior point z₀ equals the average of u over any circle centered at z₀ and contained in D, proved u…