December  2007, 19(4): 675-690. doi: 10.3934/dcds.2007.19.675

Action functionals that attain regular minima in presence of energy gaps

1. 

EPFL, Chaire d’Analyse Mathématiques et Applications, CH-1015 Lausanne, Switzerland

Received  October 2006 Revised  March 2007 Published  September 2007

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.

Citation: Alessandro Ferriero. Action functionals that attain regular minima in presence of energy gaps. Discrete and Continuous Dynamical Systems, 2007, 19 (4) : 675-690. doi: 10.3934/dcds.2007.19.675
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