For the solution of the Poisson problem with an L∞ right hand side
\begin{cases} -\Delta u(x) = f (x) & {\rm in}\ D, \\ u=0 & {\rm on}\ \partial D \end{cases}
we derive an optimal estimate of the form
\|u\|_\infty\leq \|f\|_\infty \sigma_D(\|f\|_1/\|f\|_\infty),
where σ
D is a modulus of continuity defined in the interval [0, |
D|] and depends only on the domain
D. The inequality is optimal for any domain
D and for any values of
$\|f\|_1$
and
$\|f\|_\infty .$
We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
where
B is a ball and |
B| = |
D|. Using this optimality property of σ
D, we derive Brezis–Galloute–Wainger type inequalities on the
L∞ norm of
u in terms of the
L1 and
L∞ norms of
f. As an application we derive
L∞ −
L1 estimates on the
k-th Laplace eigenfunction of the domain
D.