【online】The Lax-Oleinik representation in non-compact setting

Here we think of the case where U : [0, +∞[×M → R, with M is a manifold.

If M is compact, as has been known for a long time, the maximum principle yields uniqueness for a given initial condition U|{0}×M. This in turn implies the representation by a Lax-Oleinik type formula.

When M is not compact, the global maximum principle does not immediately hold.

Hitoshi Ishii and his coworkers obtained results about 10 years ago under some restrictions when M = Rn. Basically the restrictions are about controlled growth at infinity.

We will explain that under the hypothesis that H is Tonelli, all continuous solutions of the evolution Hamilton-Jacobi equation above satisfy the Lax-Oleinik representation even for non-compact M. This of course will imply uniqueness for a given initial condition.

Moreover, we will also show that if any pointwise finite U is given by the Lax-Oleinik representation is automatically continuous and therefore a viscosity solution.