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 {-Δu(x)=f(x) in D, u=0 on D we derive an optimal estimate of the form ||u||∞ ≤||f||∞ σD(||f||1||f||∞) where σD 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||∞ We also show that σD(t) ≤ σB(t), for t ϵ [0,|D|] where B is a ball and |B| = |D|. Using this optimality property of σD, 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.