Some new PDE methods for weak KAM theory

We discuss a new approximate variational principle for weak KAM theory. The advantage of this approach is that we build both a minimizing measure and a solution of the generalized eikonal equation at the same time. Furthermore the approximations are smooth, and so we can derive some interesting formulas upon differentiating the Euler-Lagrange equation. Our method is inspired by the ”calculus of variations in the sup-norm” ideas of Aronsson, Jensen, Barron and others. Received: 30 November 2001 / Accepted: 23 January 2002 / Published online: 5 September 2002 RID="*" ID="*" Supported in part by NSF Grant DMS-0070480 and by the Miller Institute for Basic Research in Science, UC Berkeley