On a conjecture of De Giorgi related to homogenization

Abstract

For a periodic vector field $\mathbf{F}$, let ${\mathbf{X}}^{\epsilon }$ solve the dynamical system \begin{equation*} \frac{d{\bf X}^ε}{dt} = {\bf F}\left(\frac {{\bf X}^ε}ε\right) . \end{equation*} In \cite{DeGiorgi} Ennio De Giorgi enquiers whether from the existence of the limit ${\mathbf{X}}^0(t):=\underset{\epsilon \to 0}{\lim }{\mathbf{X}}^{\epsilon }(t)$ one can conclude that $\frac{d{\mathbf{X}}^0}{dt}=constant$. Our main result settles this conjecture under fairly general assumptions on $\mathbf{F}$, which may also depend on $t$-variable. Once the above problem is solved, one can apply the result to the transport equation, in a standard way. This is also touched upon in the text to follow.