On a conjecture of De Giorgi related to homogenization

Abstract

For a periodic vector field F, let X^varepsilon solve the dynamical system dXεdt=FXεε.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\frac{{\\hbox {d}}{\\hbox {X}}^{\\varepsilon }}{{\\hbox {d}}t} = {{F}}\\left( \\frac{{X}^{\\varepsilon }}{\\varepsilon }\\right) . \\end{aligned}$$\\end{document}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 nolimits _{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.