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-infinity right hand side {-Delta u(x) = f(x) in D, u = 0 on partial derivative D we derive an optimal estimate of the form parallel to u parallel to infinity <= parallel to f parallel to(infinity)sigma(D)(parallel to f parallel to(1)/parallel to f parallel to(infinity)), where sigma(D) is a modulus of continuity defined in the interval [0, vertical bar D vertical bar] and depends only on the domain D. The inequality is optimal for any domain D and for any values of parallel to f parallel to(1) and parallel to f parallel to(infinity). We also show that sigma(D)(t) <= sigma(B)(t), for t is an element of [0, vertical bar D vertical bar], where B is a ball and vertical bar B vertical bar = vertical bar D. Using this optimality property of sigma(D), we derive Brezis-Galloute-Wainger type inequalities on the L-infinity norm of u in terms of the L-1 and L-infinity norms of f. As an application we derive L-infinity - L-1 estimates on the k-th Laplace eigenfunction of the domain D.