Abstract
A comparison between some relaxation methods of an integral functional is carried out. The following relaxed functionals of the variational integral I(Ω, u)=\(\mathop \smallint \limits_\Omega f(x,Du)\):
are introduced. It is proved, by means of examples, that in general such functionals are different even if Ω is a regular bounded open set and criteria for identity on the whole L1 (Ω) are proved. If f does not depend on x it is proved that Î and Ī agree if Ω has Lipschitz boundary and an integral representation formula for their common values on BV(Ω) is proved. Similar results and comparison ones with Î and Ī are proved also for other kinds of relaxed functionals of I.
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Esposito, A.C., De Arcangelis, R. Comparison results for some types of relaxation of variational integral functionals. Annali di Matematica pura ed applicata 164, 155–193 (1993). https://doi.org/10.1007/BF01759320
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DOI: https://doi.org/10.1007/BF01759320