86 2. FOUR IMPORTANT LINEAR PDE 6. Let U be a bounded, open subset of Rn. Prove that there exists a constant C, depending only on U, such that max ¯ U |u| ≤ C(max ∂U |g| + max ¯ U |f|) whenever u is a smooth solution of −Δu = f in U u = g on ∂U. (Hint: −Δ(u + |x|2 2n λ) ≤ 0, for λ := max ¯ U |f|.) 7. Use Poisson’s formula for the ball to prove rn−2 r − |x| (r + |x|)n−1 u(0) ≤ u(x) ≤ rn−2 r + |x| (r − |x|)n−1 u(0) whenever u is positive and harmonic in B0(0,r). This is an explicit form of Harnack’s inequality. 8. Prove Theorem 15 in §2.2.4. (Hint: Since u ≡ 1 solves (44) for g ≡ 1, the theory automatically implies ∂B(0,1) K(x, y) dS(y) = 1 for each x ∈ B0(0, 1).) 9. Let u be the solution of Δu = 0 in R+n u = g on ∂R+n given by Poisson’s formula for the half-space. Assume g is bounded and g(x) = |x| for x ∈ ∂R+, n |x| ≤ 1. Show Du is not bounded near x = 0. (Hint: Estimate u(λen)−u(0) λ .) 10. (Reflection principle) (a) Let U + denote the open half-ball {x ∈ Rn | |x| 1, xn 0}. Assume u ∈ C2(U +) is harmonic in U +, with u = 0 on ∂U + ∩ {xn = 0}. Set v(x) := u(x) if xn ≥ 0 −u(x1, . . . , xn−1, −xn) if xn 0 for x ∈ U = B0(0, 1). Prove v ∈ C2(U) and thus v is harmonic within U.

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