Action functionals that attain regular minima in presence of energy gaps

We present three simple regular one-dimensional variational problems that present the Lavrentiev gap phenomenon, i.e., inf$\{\int_a^b L(t,x,\dot x): x\in W_0^{1,1}(a,b)\} $< inf$\{\int_a^bL(t,x,\dot x): x\in W_0^{1,\infty}(a,b)\}$ (where $ W_0^{1,p}(a,b)$ denote the usual Sobolev spaces with zero boundary conditions), in which in the first example the two infima are actually minima, in the second example the infimum in $ W_0^{1,\infty}(a,b)$ is attained while the infimum in $ W_0^{1,1}(a,b)$ is not, and in the third example both infimum are not attained. We discuss also how to construct energies with a gap between any space and energies with multi-gaps.