26 2. FOUR IMPORTANT LINEAR PDE and consequently, using Green’s formulas from §C.2, we compute φ (r) = − ∂B(x,r) Du(y) · y − x r dS(y) = − ∂B(x,r) ∂u ∂ν dS(y) = r n − B(x,r) Δu(y) dy = 0. Hence φ is constant, and so φ(r) = lim t→0 φ(t) = lim t→0 − ∂B(x,t) u(y) dS(y) = u(x). 2. Observe next that our employing polar coordinates, as in §C.3, gives B(x,r) u dy = r 0 ∂B(x,s) u dS ds = u(x) r 0 nα(n)sn−1ds = α(n)rnu(x). THEOREM 3 (Converse to mean-value property). If u ∈ C2(U) satisfies u(x) = − ∂B(x,r) u dS for each ball B(x, r) ⊂ U, then u is harmonic. Proof. If Δu ≡ 0, there exists some ball B(x, r) ⊂ U such that, say, Δu 0 within B(x, r). But then for φ as above, 0 = φ (r) = r n − B(x,r) Δu(y) dy 0, a contradiction. 2.2.3. Properties of harmonic functions. We now present a sequence of interesting deductions about harmonic functions, all based upon the mean-value formulas. Assume for the following that U ⊂ Rn is open and bounded.

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