TITLE: A calculus problem on the poisson equation
QUESTION [6 upvotes]: Let $u(x, y)$ be a function that equals its Laplacian ($u \equiv \Delta u$) on the unit disk $\mathbb{D}$ ($x^2 + y^2 \leq 1$) and $u \mid_{\partial \mathbb{D}} \equiv 0$ ($\partial \mathbb{D}$ is the circle $x^2 + y^2 = 1$). Please prove that $u \equiv 0$ on $\mathbb{D}$.
Note: This problem is taken from the Ph. D. qualification test of Peking University, 2019.
My failed attempts:

Construct a function $f$ related to $u$ such that $\Delta f = 0$, and apply the Maximum Principle of harmonic functions.

Consider the integral $F(r) = \int_0^{2\pi} u(x + r\cos\theta, y + r\sin\theta) \, d\theta$ and get $rF(r) = rF''(r) + F'(r)$ but fail to continue.

Consider the Fourier transform of $u$.


This problem might be easy for the scholars who have a profound expertise in functional analysis and PDE (However, I do not). Would you please offer several hints or point out the insights behind this problem? 50 reputation will be awarded if you can prove the case where $u \in C^\infty(\mathbb{R}^2)$. I will keep my promise like How to calculate the Fourier transform of the Kaiser-Bessel window?.

REPLY [3 votes]: Unless I am grossly mistaken, this should follow from the Divergence theorem:
$$\iint_{\mathbb{D}}(\|\nabla u\|^2 + u^2)dA = \iint_{\mathbb{D}}(\nabla u\cdot\nabla u + u(\nabla\cdot\nabla u))dA  = $$
$$ = \iint_{\mathbb{D}}\nabla\cdot(u\nabla u)dA = \int_{\partial{\mathbb{D}}}u\nabla u\cdot\boldsymbol{n}dS = 0$$
and so $u$ is identically zero.