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- W2022665642 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)= $$mathop smallint limits_Omega f(x,Du)$$ : $$begin{gathered} hat I(Omega ,u) = inf { mathop {lim inf }limits_h I(Omega ,u_h ),u_h in C^1 (R^n ),u_h to u in L^1 (Omega )} u in L^1 (Omega ), hfill bar I(Omega ,u) = inf { mathop {lim inf }limits_h I(Omega ,u_h ),u_h in C^1 (Omega ),u_h to u in L_{loc}^{^1 } (Omega )} u in L_{loc}^{^1 } (Omega ) hfill end{gathered} $$ 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." @default.
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- W2022665642 date "1993-12-01" @default.
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- W2022665642 title "Comparison results for some types of relaxation of variational integral functionals" @default.
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- W2022665642 doi "https://doi.org/10.1007/bf01759320" @default.
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