Non-occurrence of gap for one-dimensional non-autonomous functionals

Let $$F(y){{:}{=}}\int \nolimits _t^TL(s, y(s), y'(s))\,ds$$ be a positive functional defined on the space $$W^{1,p}([t,T]; {{\mathbb {R}}}^n)$$ ( $$p\ge 1$$ ) of Sobolev functions with, possibly, one or both end point conditions. It is important, especially for applications, to able approximate infimum F with values along sequence Lipschitz satisfying same boundary condition(s). Sometimes this not possible, i.e., so called Lavrentiev phenomenon occurs. This case seemingly innocent Manià’s Lagrangian $$L(s,y,y')=(y^3-s)^2(y')^6$$ and data $$y(0)=0, y(1)=1$$ ; nevertheless in situation does occur just condition $$y(1)=1$$ . The paper focuses about different set conditions that are needed avoid problems depending number considered. Under minimal assumptions (possibly) extended value, Lagrangian, we ensure non-occurrence condition, thus extending milestone result by Alberti Serra Cassano non-autonomous case. We then introduce an additional hypothesis, satisfied when bounded sets, order dealing conditions; gives some new light even autonomous