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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