Action functionals that attain regular minima in presence of energy gaps

Pages: 675 - 690, Volume 19, Issue 4, December 2007

doi:10.3934/dcds.2007.19.675       Abstract        Full Text (192.6K)       Related Articles

Alessandro Ferriero - EPFL, Chaire d’Analyse Mathématiques et Applications, CH-1015 Lausanne, Switzerland (email)

Abstract: 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.

Keywords:  One-dimensional variational problems, Lavrentiev phenomenon, singular phenomena, regularity of minimizers.
Mathematics Subject Classification:  Primary: 49J30; secondary: 49J45, 49N60.

Received: October 2006;      Revised: March 2007;      Published: September 2007.