I was studying Stein and Shakarchi's book on Fourier analysis, and came across this theorem.
Let $f$ be an integrable function defined on the unit circle. Then the function $u$ defined in the unit disc by the Poisson integral $$u(r, \theta)=(f* P_r)(\theta)$$ has the following properties:(i) $u$ has two continuous derivatives in the unit disc and satisfies $\Delta u=0$.
(ii) If $\theta$ is any point of continuity of $f$, then $\lim_{r\to 1} u(r, \theta)=f(\theta)$. If $f$ is continuous everywhere, then this limit is uniform.
(iii) If $f$ is continuous, then $u(r,\theta)$ is the unique solution to the steady-state heat equation in the disc which satisfies conditions (i) and (ii).
Remark: If uniform convergence is replaced by pointwise convergence, uniqueness fails.
This remark made me think about the condition that $u(r, \theta)$ converges uniformly to $f(\theta)$ as $r\to 1$. Is this condition a 'physically trivial' condition? What is its meaning?
Also, is there a theorem that guarantees this uniform convergence with additional conditions on pointwise convergence (something like Dini's Theorem, but monotone would not be ideal)? I would like to have such a theorem to understand the meaning of this uniform convergence better.
It is a very basic condition, I wish the book had explained this. This condition is just equivalent to the (joint) continuity of $u$ on the unit disc.