*Published Paper*

**Inserted:** 19 apr 2007

**Last Updated:** 15 jun 2010

**Journal:** Discrete Contin. Dyn. Syst.

**Year:** 2007

**Abstract:**

We present three simple regular one-dimensional variational problems that present the Lavrentiev gap phenomenon, i.e.
$$\inf\left\{\int_{a}^{b} L(t,x,\dot x):
x\in**W**_{0}^{{1,1}}(a,b)\right\}<
\inf\left\{\int_{a}^{bL}(t,x,\dot x):
x\in**W**_{0}^{{1,\infty}}(a,b)\right\},$$
(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 meanwhile 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 gap between any space and energies with multi-gaps.

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