Lindelof Maximum Principle

Theorem 1. Suppose u is bounded and harmonic in a bounded open connected set W. Suppose lim sup_{z→p∈∂W} u(z) ≤ M except for a countable set {q_{j} ∈ ∂W }. Then u(z) ≤ M.

Proof. Choose ϵ_{j} > 0 so that ∑_{1}^{∞} ϵ_{j} < ∞. Let d be the diameter of W. Let a ∈ W and let c be the distance

from a to ∂W. Then

w(z) = ∑ ϵ_{j} log \frac{|z − q_{j}|}{d}

converges uniformly on compact subsets of W to a harmonic function. Hence v(z) = u(z) + w(z) is harmonic on W. Also v(z) ≤ M. Let B = sup\{v(z) : z ∈ W. Then there is a point b ∈ \overline{W} and sequence z_{j} → b, z_{j} ∈ W. If v is not constant, b ∈ ∂W. Also b ≠ q_{j} for any j, so it must be that B ≤ M. Now if we fix a, we see that v(a) = u(a) + w(a) ≤ M and since we can choose ∑ ϵ_{j} to be arbitrarily small, we see that B ≤ M.

Remark 1. It’s not obvious how to modify this for a set of measure 0.