On the L∞-maximization of the solution of Poisson's equation: Brezis–Gallouet–Wainger type inequalities and applications

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.