On a conjecture of De Giorgi related to homogenization

TL;DR

This paper proves a conjecture by De Giorgi regarding the behavior of solutions to a homogenization problem involving periodic vector fields, showing that the limit of the dynamical system's solutions has a constant derivative under broad conditions.

Contribution

The paper establishes the validity of De Giorgi's conjecture for a wide class of vector fields, including those depending on time, advancing understanding in homogenization theory.

Findings

Confirmed the limit's derivative is constant under general conditions

Applied the result to the transport equation

Extended the conjecture's validity to time-dependent fields

Abstract

For a periodic vector field $\bf F$, let ${\bf X}^\epsilon$ solve the dynamical system \begin{equation*} \frac{d{\bf X}^\epsilon}{dt} = {\bf F}\left(\frac {{\bf X}^\epsilon}\epsilon\right) . \end{equation*} In \cite{DeGiorgi} Ennio De Giorgi enquiers whether from the existence of the limit ${\bf X}^0(t):=\lim\limits_{\epsilon\to 0}{\bf X}^\epsilon(t)$ one can conclude that $ \frac{d{\bf X}^0}{dt}= constant$. Our main result settles this conjecture under fairly general assumptions on $\bf F$, 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.