I am studying Dirichlet's problem on the half superior plane with the book Mesures, Intégration, Convolution et Transformée de Fourier des fonctions from El Haj Laamri.
The problem is the following:
We are looking for $u\in C^2(\mathbb R\times ]0,+\infty[)$ satisfying
$$(\mathcal D)\quad \begin{cases} \displaystyle\frac{\partial ^2u}{\partial x^2}+\frac{\partial ^2u}{\partial y^2}=0 \\ u(x,0)=f(x) \\ \displaystyle\sup_{y\geqslant 0}\int_{-\infty}^{+\infty}\vert{u(x,y)}\vert\mathrm d x=0 \end{cases}$$
where $f\in L^1(\mathbb R)$. We also assume that for all $y>0$ : $u(\cdot,y)$, $\displaystyle\frac{\partial u}{\partial x}(\cdot ,y)$, $\displaystyle\frac{\partial^2 u}{\partial x^2}(\cdot ,y)$ and $\displaystyle\frac{\partial^2 u}{\partial y^2}(\cdot ,y)$ are integrables on $\mathbb R$.
Thanks to this other question, I know that a function $u$ satisfying the third condition
$$\displaystyle\sup_{y\geqslant 0}\int_{-\infty}^{+\infty}\vert{u(x,y)}\vert\mathrm d x=0$$
must be null everywhere.
This is why I do not understand this condition.
Is there an error in the book?
Should it be:
$\displaystyle\limsup_{y\geqslant 0}\int_{-\infty}^{+\infty}\vert{u(x,y)}\vert\mathrm d x=0$?
$\displaystyle\sup_{y\geqslant 0}\int_{-\infty}^{+\infty}\vert{u(x,y)}\vert\mathrm d x<+\infty$?