On one method of improving weakly converging sequence of gradients

Abstract

Let Ω⊆Rⁿ be a bounded open domain, F⊆Omega¯ be a closed subset and let {u_k}_k∈N be a bounded sequence in Sobolew space where 1≤p<∞, converging weakly to u∈W^1,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 μ_f. We discuss the possibility to modify the sequence {u_k}_k∈N in such a way that the new sequence {w_k}_k∈N is still bounded in W^1,p(Ω, R^m), converges weakly to u, each sequence of measures {f(∇w_k)(1+|∇w_k|^p) dx}_k∈N also converges weakly * to μ_f, where f∈ℛ, but additionally the new sequence satisfies the condition “w_k=u” on F. Our results are applied to the minimization problems in the Calculus of Variations.