# On a conjecture of De Giorgi related to homogenization

Annali di Matematica Pura ed Applicata 1923 -

pp 1–17

First Online: 05 May 2017Received: 26 August 2015Accepted: 17 April 2017DOI: 10.1007-s10231-017-0659-y

Cite this article as: Karakhanyan, A.L. & Shahgholian, H. Annali di Matematica 2017. doi:10.1007-s10231-017-0659-y

Abstract

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 } ight . \end{aligned}In Set Valued Anal 21–2:175–182, 1994 Ennio De Giorgi enquiers whether from the existence of the limit \ X^0t:=\lim olimits {\varepsilon ightarrow 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.KeywordsDynamical system ODE Transport Homogenization Convergence rate H. Shahgholian was supported by Swedish Research Council. A. Karakhanyan was partly supported by EPSRC Grant. We thank Michael Benedicks for his insightful comments on the dynamical system issues of the current note, and Björn Engquist for bringing to our attention the paper 8.

Mathematics Subject Classification34C29 37A10 65L70 74Q10

Author: Aram L. Karakhanyan - Henrik Shahgholian