2.2. LAPLACE’S EQUATION 31 Proof. Since Φ(x) → 0 as |x| → ∞ for n ≥ 3, ˜(x) u := Rn Φ(x − y)f(y) dy is a bounded solution of −Δu = f in Rn. If u is another solution, w := u− ˜ u is constant, according to Liouville’s Theorem. Remark. If n = 2, Φ(x) = − 1 2π log |x| is unbounded as |x| → ∞ and so may be R2 Φ(x − y)f(y) dy. e. Analyticity. We refine Theorem 6: THEOREM 10 (Analyticity). Assume u is harmonic in U. Then u is analytic in U. Proof. 1. Fix any point x0 ∈ U. We must show u can be represented by a convergent power series in some neighborhood of x0. Let r := 1 4 dist(x0,∂U). Then M := 1 α(n)rn u L1(B(x0,2r)) ∞. 2. Since B(x, r) ⊂ B(x0, 2r) ⊂ U for each x ∈ B(x0,r), Theorem 7 provides the bound Dαu L∞(B(x0,r)) ≤ M 2n+1n r |α| |α||α|. Now kk k! ek for all positive integers k, and hence |α||α| ≤ e|α||α|! for all multiindices α. Furthermore, the Multinomial Theorem (§1.5) implies nk = (1 + · · · + 1)k = |α|=k |α|! α! , whence |α|! ≤ n|α|α!. Combining the previous inequalities yields the estimate (22) Dαu L∞(B(x0,r)) ≤ M 2n+1n2e r |α| α!. 3. The Taylor series for u at x0 is α Dαu(x0) α! (x − x0)α,

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