We find explicitly the Bellman function for the dyadic maximal operator on L^p as the solution of a Bellman partial differential equation of Monge-Ampère type. This function has been previously found by A. Melas (2005) in a different way, but it is our partial differential equation-based approach that is of principal interest here. Clear and replicable, it holds promise as a unifying template for past and current Bellman function investigations. To cite this article: L. Slavin et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).