Calculus of Variations and Geometric Measure Theory

A. Ferriero

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.


Download: