Partial Differential Equations

1.2 The Laplace Equation 11
Exercise 1.2.8. Vary the proof of the mean value property for harmonic functions to show that, for
n ≥ 3, a solution of the boundary value problem
{ −∆u = f on B(0; r),
satisfies the integral formula
∫
u(0) =
r n
vol ∂B(0;r)
∂B(0;r)
u = g on ∂B(0; r)
g(y) dS ∂B(0;r) (y) + 1
1
n(n−2) vol B(0;r)
∫
B(0;r)
( 1
|y| n−2 − 1 r 2 )
f(y)dy.
Theorem 1.2.9 (Maximum principle). Let U ⊆ R n be open and bounded. For every harmonic function
u: U → R which can be extended continuously to U, we have
(i) max x∈U
u(x) = max x∈∂U u(x).
(ii) Let U be connected, and let x 0 ∈ U with u(x 0 ) = max x∈U
u(x), then u is constant on U.
Proof. Idea: Use the mean value property to prove (ii). Then (i) follows.
Since U is compact, the maxima exist, and (i) follows from (ii). To show (ii), we set M := u(x 0 ). If
x ∈ U and u(x) = M, then we choose a ball B(x; r) whose closure is contained in U. Then, Lemma 1.2.7
gives
∫
1
M =
u(y)dy ≤ M,
vol B(x; r) B(x;r)
and we have u(y) = M for all y ∈ B(x; r). Thus, u −1 (M) ⊆ U is open. But, u −1 (M) also is closed since
u is continuous. Since U is assumed to be connected and u −1 (M) is nonempty, u has to be constant and
equal to M.
Corollary 1.2.10. Let U ⊆ R n be open and bounded. If two harmonic functions can be continuously
extended onto U, and if they coincide on ∂U, then they are equal.
Proof. This immediately follows from the Maximum Principle 1.2.9 for harmonic functions.
Let U ⊆ R n be open and bounded. Then, by Corollary 1.2.10, there is at most one harmonic function
ϕ x on U for x ∈ U which can continuously be extended onto U, and which satisfies
ϕ x (y) = Φ(y − x)
∀y ∈ ∂U.
In case that all ϕ x exist (one can prove the existence of the ϕ x , but one needs methods which are not at
our disposal yet), the function
G: {(x, y) ∈ U × U | x ≠ y} → R,
(x, y) ↦→ Φ(y − x) − ϕ x (y)
is called Green’s function for the Laplace equation on U. Quite general, the name Green’s function is
used for arbitrary open subsets U ⊆ R n as soon as one has chosen a family (ϕ x ) x∈U of harmonic functions
with ϕ x (y) = Φ(x − y) for y ∈ ∂U.
Lemma 1.2.11. Let U ⊆ R n be an open and bounded subset with smooth boundary. For u ∈ C 2 (U) and
x ∈ U, we have the formula
∫ (
u(x) = − u(y) ∂Φ
)
∫
(y − x) − Φ(y − x)∂u
∂U ∂ν ∂ν (y) dS ∂U (y) − Φ(y − x)∆u(y) dy.
U