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Does there exist a (onedimensional) integral functional of calculus of variations $$ F(y)=\int_a^b f(t,y(t),y'(t))\,dt
 $$ such that not only $$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])}F(y) $$ (that is, it shows the Lavrentiev phenomenon), but actually both the two $\inf$ are $\min$?

That is, both the two minimization problems have a solution but they are different?

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    $\begingroup$ And exactly what $f$ are allowed in this game? $\endgroup$
    – fedja
    Commented Dec 13, 2017 at 4:05
  • $\begingroup$ I think that if $f$ can be only Borel, your idea in problem mathoverflow.net/questions/288336/… with the function $f(t,y,ξ)$ to be 0 if y^3=|t| and F(y) otherwise, should work. $\endgroup$
    – Carlo Mantegazza
    Commented Dec 13, 2017 at 11:23
  • $\begingroup$ An example with $f$ Caratheodory/continuous (even polynomial) would be very interesting... the simplest example of Lav-phen is with a polynomial $f$. $\endgroup$
    – Carlo Mantegazza
    Commented Dec 13, 2017 at 11:26
  • $\begingroup$ I think that the paper of A. Ferriero (Action functionals that attain regular minima in presence of energy gaps, Discr. Cont. Dyn. Syst 2007, link) may help. $\endgroup$
    – Simone Vincini
    Commented Mar 25, 2019 at 6:51

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