28 2. FOUR IMPORTANT LINEAR PDE THEOREM 5 (Uniqueness). Let g ∈ C(∂U), f ∈ C(U). Then there exists at most one solution u ∈ C2(U) ∩ C( ¯ U ) of the boundary-value problem (17) −Δu = f in U u = g on ∂U. Proof. If u and ˜ u both satisfy (17), apply Theorem 4 to the harmonic functions w := ±(u − ˜). u b. Regularity. Next we prove that if u ∈ C2 is harmonic, then necessarily u ∈ C∞. Thus harmonic functions are automatically infinitely differentiable. This sort of assertion is called a regularity theorem. The interesting point is that the algebraic structure of Laplace’s equation Δu = ∑n i=1 uxixi = 0 leads to the analytic deduction that all the partial derivatives of u exist, even those which do not appear in the PDE. THEOREM 6 (Smoothness). If u ∈ C(U) satisfies the mean-value prop- erty (16) for each ball B(x, r) ⊂ U, then u ∈ C∞(U). Note carefully that u may not be smooth, or even continuous, up to ∂U. Proof. Let η be a standard mollifier, as described in §C.5, and recall that η is a radial function. Set uε := ηε ∗ u in Uε = {x ∈ U | dist(x, ∂U) ε}. As shown in §C.5, uε ∈ C∞(Uε). We will prove u is smooth by demonstrating that in fact u ≡ uε on Uε. Indeed if x ∈ Uε, then uε(x) = U ηε(x − y)u(y) dy = 1 εn B(x,ε) η |x − y| ε u(y) dy = 1 εn ε 0 η r ε ∂B(x,r) u dS dr = 1 εn u(x) ε 0 η r ε nα(n)rn−1dr by (16) = u(x) B(0,ε) ηε dy = u(x). Thus uε ≡ u in Uε, and so u ∈ C∞(Uε) for each ε 0.
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