On the $L^{\infty }-$maximization of the solution of Poisson's equation: Brezis-Gallouet-Wainger type inequalities and applications

Abstract

For the solution of the Poisson problem with an $L^{\infty }$ right hand side \begin{equation*} \begin{cases} -Δu(x) = f (x) & \mbox{in } D, u=0 & \mbox{on } \partial D, \end{cases} \end{equation*} we derive an optimal estimate of the form $$\Vert u{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }{\sigma }_D(\Vert f{\Vert }_1/\Vert f{\Vert }_{\infty }),$$ where ${\sigma }_D$ is a modulus of continuity defined in the interval $[0,|D|]$ and depends only on the domain $D$. In the case when $f\ge 0$ in $D$ the inequality is optimal for any domain and for any values of $\Vert f{\Vert }_1$ and $\Vert f{\Vert }_{\infty }.$ We also show that $${\sigma }_D(t)\le {\sigma }_B(t),\text{ for }t\in [0,|D|],$$ where $B$ is a ball and $|B|=|D|$. Using this optimality property of $\sigma ,$ we derive Brezis-Galloute-Wainger type inequalities on the $L^{\infty }$ norm of $u$ in terms of the $L^1$ and $L^{\infty }$ norms of $f.$ The estimates have explicit coefficients depending on the space dimension $n$ and turn to equality for a specific choice of $u$ when the domain $D$ is a ball. As an application we derive $L^{\infty }-L^1$ estimates on the $k-$th Laplace eigenfunction of the domain $D.$

10 pages