ABSTRACT
In this section, we establish basic elliptic regularity estimates for harmonic functions and solve the Dirichlet problem.
Proposition B.1.1 Let Ω be an open set of RN , N ≥ 1. Let u ∈ L1loc(Ω) be a harmonic function. Then, u ∈ C∞(Ω), and given any k ∈ NN , there exists a constant CN ,k, depending on N and k only, such that
|Dku(x0)| ≤ CN ,kr |k| ‖u‖L∞(B(x0,r)), (B.1)
for every ball B(x0, r)⊂ Ω. Proof. Let u ∈ L1loc(Ω) be a harmonic function. By Exercise A.1.3, u ∈ C2(Ω). We establish (B.1) by induction on |k|. The case |k| = 0 is a straightforward consequence of the mean-value formula (A.10). Now differentiate the meanvalue formula (A.10) with respect to x i. Then, v = ∂ u/∂ x i also satisfies the
of
mean-value formula. By Exercise A.1.3, v is harmonic. In particular, u ∈ C3(Ω). Iterating this argument, we deduce that u ∈ C∞(Ω). In addition, ∂ u∂ x i (x0)
∂ u
∂ x i d y
=
|B1|
r
uni dσ
≤ C
r ‖u‖L∞(∂ B(x0,r/2)). (B.2)
This implies (B.1) for |k| = 1. Now take n ≥ 1 and assume that (B.1.1) holds for all |k| ≤ n. Take k′ ∈ NN such that |k′|= n+ 1. Then,
Dk ′ u=
∂
∂ x i Dku,
for some i ∈ {1, . . . , N} and k ∈ NN such that |k| = n. By (B.2) applied to v = Dku, we have
|Dk′u(x0)| ≤ Cr ‖D ku‖L∞(∂ B(x0,r/2)). (B.3)
Now, if x ∈ ∂ B(x0, r/2), then B(x , r/2) ⊂ B(x0, r) ⊂ Ω. Using the induction hypothesis, we deduce that
This together with (B.3) yields the desired result. ƒ Exercise B.1.1 Let u be a harmonic function in Ω⊂ RN , N ≥ 1. Prove that for every ball B(x0, r)⊂ Ω and every k ∈ NN ,
|Dku(x0)| ≤ CN ,krN+|k|‖u‖L1(B(x0,r)), where CN ,0 = 1/|B1| and for |k| ≥ 1,
CN ,k ≤ 2N+1N |k||k| |B1| .
