Abstract
We present a constructive proof of the fact, that for any subset 𝒜 ⊆ ℝm and a countable family ℱ of bounded functions f : 𝒜 → ℝ there exists a compactification 𝒜′ ⊂ ℓ2 of 𝒜 such that every function f ∈ ℱ possesses a continuous extension to a function f̅ : 𝒜′→ℝ. However related to some classical theorems, our result is direct and hence applicable in Calculus of Variations. Our construction is then used to represent limits of weakly convergent sequences {f(uν)} via methods related to DiPerna-Majda measures. In particular, as our main application, we generalise the Representation Theorem from the Calculus of Variations due to Kałamajska.
Highlights
The behaviour of a composition of a weakly convergent sequence with a nonlinear mapping is essentially non-trivial
Taking Ω = (0, 1)n and knowing knowing that uν converges weakly to u in L1(Ω) we are not able to say much about the weak limit in L1(Ω) of the sequence f, where f : R → R is of a linear growth
The proof of the Representation Theorem 2.11 exploits a distance function on γAi, while one of other assertions of the statement uses a support of the certain DiPerna-Majda measure defined on γAi
Summary
The behaviour of a composition of a weakly convergent sequence with a nonlinear mapping is essentially non-trivial. The theorem shows a representation formula for the weak- limit for the sequences of compositions {f (uν)dμ}, where f : Rm → R is a continuous function on every set Ai, i = 1, . The proof of the Representation Theorem 2.11 exploits a distance function on γAi, while one of other assertions of the statement uses a support of the certain DiPerna-Majda measure defined on γAi. For purposes of Representation Theorem 2.11, we need to know the precise shape of the compactification, construction of homeomorphic embeddings φi and insure that γAi are metric spaces. It becomes natural to ask for a theorem working in possibly general setting, so that both Representatrion Theorem 2.11 and Convergence Theorem 2.12 become its special cases Such a generalisation may contribute to both Calculus of Variations and Set-valued Analysis.
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