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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