On one method of improving weakly converging sequence of gradients

摘要

Let Ω is contained in R~n be a bounded open domain, F is contained in Ω be a closed subset and let {u_k}_(k∈N) be a bounded sequence in Sobolew space where L ≤ p < ∞, converging weakly to u ∈ W~(l,p)(Ω, R~m). Let R be a given complete separable ring of continuous functions on R~(m×n) and assume that for each f ∈ R the sequence of compositions {f(▽u_k)(1 + |▽u_k|~p)dx}_(k∈N) embedded into the space of measures on Ω converges weakly * to some measure μf. We discuss the possibility to modify thernsequence {u_k}_(k∈N) in such a way that tne new sequence {w_k}_(k∈N) is still bounded in W~(l,p)(Ω, R~m), converges weakly to u,rneach sequence of measures {f(▽w_k)(1 + |▽w_k|~p)dx}_(k∈N) also converges weakly * to μ_f, where f ∈R, but additionally thernnew sequence satisfies the condition "w_k = u" on F. Our results are applied to the minimization problems in the Calculus ofrnVariations.