**Question:**
Let us consider the Poisson problem on the square with constant source $1$
$$
\begin{cases}
- \Delta u &= 1, \qquad \text{ in } (0,1)^n \\\\
u &= 0, \qquad \text{ on } \partial (0,1)^n
\end{cases}
$$
By symmetry the maximum is attained in $x_0:=(1/2,\dots,1/2)$. Does there exist a formula or precise estimate for
$$
\max u((0,1)^n) = u(x_0) \quad ?
$$
How does this value scale in $n$? Can one recover some bound $o(1)$, for $n\to \infty$?

**Background:**
It is easy by comparison with $1/8-(x-x_0)^2/(2n)$ and the maximum principle to deduce the bound
$$
u(1/2, \dots, 1/2 ) \leq 1/8 ,
$$
which is also sharp for $n=1$, where $u(x) = 1/2 \; x (1-x)$ is the solution. By the same method one finds for the solution of the Poisson problem on the unit ball the bound $1/(2n)$, which is the second part of the question.

**Strategy so far:** By the product-ansatz one can deduce an explicit solution for $u$, given by
$$
u(x) = \frac{4^n}{\pi^{n+2}}\sum_{\alpha\in (2 \mathbb{N}+1)^n} \frac{1}{\sum_{i=1}^n \alpha_i^2} \prod_{j=1}^n \frac{\sin(\pi \alpha_j x_j)}{\alpha_j} .
$$
Hereby, I denote with $2\mathbb{N}+1$ the odd integers starting with $1$, hence $\alpha\in (2\mathbb{N}+1)^n$ is a multiindex with odd entries. In principle, setting $x=(1/2,\dots,1/2)$ should give the solution.

How does one calculate the resulting alternating series or find estimates recovering at least the scaling in $n$?

**Numerical results:** For small $n$ from series representation:

```
n=1: 0.125
n=2: 0.0737
n=3: 0.0562
n=4: 0.0473
```

**Probabilistic interpretation of the solution:** $u(x)$ is the mean hitting time of a Brownian particle starting in $x$, which may open estimates from the stochastic side.

**One more thought wrt. volume:** It is not clear, if the comparison of unit cube and unit ball is the right comparison, since the volume of the unit ball goes to $0$ for $n\to \infty$. Maybe, one should consider the unit-volume-ball, for which I find the scaling of the maximum like $O(n^{-1/2})$. I can add details, if there is some indication in this direction.

**Bonus questions:** Is there a more general method not directly based on explicit solutions, which would lead to estimates for general domains, for instance convex, star shaped or just bounded?

What is a reasonable quantity describing the scaling behaviour in $n$? For instance: Do all reasonable unit-volume domains scale like $O(n^{-1/2})$?