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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 $O(n^{-1/2})$?

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Here is a start of a method which should give you some asymptotics for the cube. You want to find

$$\int_{t=0}^\infty p(t)^n dt$$

where $p(t)$ is the probability that a Brownian motion has not left a unit interval by time $t$ after starting at the center. As $n\to \infty$, what matters is the behavior of $p(t)$ near $0$. While the first exit time density is a little messy for two barriers, it's easy for one barrier by reflection, and you get a Lévy distribution. For small $t$, it's very unlikely that a Brownian motion would hit both endpoints of the interval, so it's a good approximation to say that the probability of exiting the interval is twice the probability of leaving in one particular direction.

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This is nice approach. In this way I obtained yet another series representation, which is not absolutely convergent. Some numerical estimates are feasible, although the scaling in $n is not obvious. – André Schlichting Apr 23 '13 at 7:46

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