On a conjecture of De Giorgi related to homogenization
Aram Karakhanyan, Henrik Shahgholian · Annali di Matematica Pura ed Applicata (1923 -) · 2017
For a periodic vector field F, let $$X^\varepsilon $$ solve the dynamical system $$\begin{aligned} \frac{{\hbox {d}}{\hbox {X}}^{\varepsilon }}{{\hbox {d}}t} = {{F}}\left( \frac{{X}^{\varepsilon }}{\varepsilon }\right) . \end{aligned}$$ In (Set Valued Anal 2(1–2):175–182, 1994) Ennio De Giorgi enquiers whether from the existence of the limit $$ X^0(t):=\lim olimits _{\varepsilon \rightarrow 0} X^\varepsilon (t)$$ one can conclude that $$ \frac{{\hbox {d}} X^0}{{\hbox {d}}t}= {\hbox {constant}}$$ . Our main result settles this conjecture under fairly general assumptions on F, which in some cases may also depend on t-variable. Once the above problem is solved, one can apply the result to the corresponding transport equation, in a standard way. This is also touched upon in the text to follow.