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Action functionals that attain regular minima in presence of energy gaps. (English) Zbl 1159.49008

Author’s summary: “We present three simple regular one-dimensional variational problems that present the Lavrentiev gap phenomenon, i.e., \[ \inf \biggl\{\int_a^bL(t,x,\dot{x})dt: x \in W_0^{1,1}(a,b) \biggr\} < \inf \biggl\{\int_a^bL(t,x,\dot{x})dt: x \in W_0^{1,\infty}(a,b) \biggr\} \] (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 infima are not attained. We discuss also how to construct energies with a gap between any space and energies with multi-gaps”.

MSC:

49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49J45 Methods involving semicontinuity and convergence; relaxation
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49N60 Regularity of solutions in optimal control
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