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.