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 $$ \inf_{y\in Lip([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])}F(y) $$$$ \inf_{y\in\operatorname{Lip}([a,b])}F(y)>\inf_{y\in W^{1,1}([a,b])}F(y) $$ (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?
