# Action functionals that attain regular minima in presence of energy gaps

created by ferrieroa on 19 Apr 2007
modified on 15 Jun 2010

[BibTeX]

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. $$\inf\left\{\intab L(t,x,\dot x): x\inW0{1,1}(a,b)\right\}< \inf\left\{\intabL(t,x,\dot x): x\inW0{1,\infty}(a,b)\right\},$$ (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.