32 2. FOUR IMPORTANT LINEAR PDE the sum taken over all multiindices. We assert this power series converges, provided (23) |x − x0| r 2n+2n3e . To verify this, let us compute for each N the remainder term: RN (x) := u(x) − N−1 k=0 |α|=k Dαu(x0)(x − x0)α α! = |α|=N Dαu(x0 + t(x − x0))(x − x0)α α! for some 0 ≤ t ≤ 1, t depending on x. We establish this formula by writing out the first N terms and the error in the Taylor expansion about 0 for the function of one variable g(t) := u(x0 + t(x − x0)), at t = 1. Employing (22), (23), we can estimate |RN (x)| ≤ M |α|=N 2n+1n2e r N r 2n+2n3e N ≤ MnN 1 (2n)N = M 2N → 0 as N → ∞. See §4.6.2 for more on analytic functions and partial differential equa- tions. f. Harnack’s inequality. Recall from §A.2 that we write V ⊂⊂ U to mean V ⊂ ¯ V ⊂ U and ¯ V is compact. THEOREM 11 (Harnack’s inequality). For each connected open set V ⊂⊂ U, there exists a positive constant C, depending only on V , such that sup V u ≤ C inf V u for all nonnegative harmonic functions u in U. Thus in particular 1 C u(y) ≤ u(x) ≤ Cu(y) for all points x, y ∈ V . These inequalities assert that the values of a non- negative harmonic function within V are all comparable: u cannot be very small (or very large) at any point of V unless u is very small (or very large) everywhere in V . The intuitive idea is that since V is a positive distance away from ∂U, there is “room for the averaging effects of Laplace’s equation to occur”.

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