30 2. FOUR IMPORTANT LINEAR PDE be a multiindex with |α| = k. Then Dαu = (Dβu)xi for some i ∈ {1, . . . , n}, |β| = k − 1. By calculations similar to those in (20), we establish that |Dαu(x0)| ≤ nk r Dβu L∞(∂B(x0, r k )) . If x ∈ ∂B(x0, r k ), then B(x, k−1 k r) ⊂ B(x0,r) ⊂ U. Thus (18), (19) for k − 1 imply |Dβu(x)| ≤ (2n+1n(k − 1))k−1 α(n) ( k−1 k r )n+k−1 u L1(B(x0,r)) . Combining the two previous estimates yields the bound (21) |Dαu(x0)| ≤ (2n+1nk)k α(n)rn+k u L1(B(x0,r)). This confirms (18), (19) for |α| = k. d. Liouville’s Theorem. We assert now that there are no nontrivial bounded harmonic functions on all of Rn. THEOREM 8 (Liouville’s Theorem). Suppose u : Rn → R is harmonic and bounded. Then u is constant. Proof. Fix x0 ∈ Rn, r 0, and apply Theorem 7 on B(x0,r): |Du(x0)| ≤ √ nC1 rn+1 u L1(B(x0,r)) ≤ √ nC1α(n) r u L∞(Rn) → 0, as r → ∞. Thus Du ≡ 0, and so u is constant. THEOREM 9 (Representation formula). Let f ∈ Cc 2(Rn), n ≥ 3. Then any bounded solution of −Δu = f in Rn has the form u(x) = Rn Φ(x − y)f(y) dy + C (x ∈ Rn) for some constant C.
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