Liouville's theorem says that a bounded harmonic function on R" is constant. To show this, assume u = C²(R") is harmonic and satisfies |u()| ≤ M for all ЄR". (a) For 70 € R", set ro = 0]. Use Corollary 9.4 (mean value for harmonic functions) Suppose CR² for n > 2. For u € C2(2), the following properties are equivalent: The function u is harmonic on 2. 1 For B(o; R), u(o) - AR-1dS, where A,, is the volume of the unit sphere in R". B(R) For B(o; R), u(o)= n = ud" A₁R" B at the centers 0 and o to show that u(0)-u(o)= [] n AnR" B(OR) Bo:R) for R>0. Note that the integrals cancel on the intersection of the two balls. < ΤΟ An n (9.41) (b) Show that vol[B (0; R)-B(70; R)] ≤ vol [B(0; R) - B(;R-)] = [R" - (R-7)"] and the same for B(70; R) - B(0; R). (c) Apply the volume estimates and the fact that || M to (9.41) to estimate that |u(0)-u(0)|≤ RT -(R- ΤΟ 2M 2 Rn Take the limit R→ ∞ to show that u(0) = u(0).