Affiliations: Institute of Mathematics, Warsaw University, Warszawa, Poland
Note: [] Address for correspondence: Agnieszka Kałamajska, Institute of Mathematics, Warsaw University, ul. Banacha 2, 02–097 Warszawa, Poland. Tel.: +48 22 55 44 553; Fax: +48 22 55 44 300; E-mail: [email protected].
Abstract: Let Ω⊆Rn be a bounded open domain, F⊆Omega¯ be a closed subset and let {uk}k∈N be a bounded sequence in Sobolew space where 1≤p<∞, converging weakly to u∈W1,p(Ω, Rm). Let ℛ be a given complete separable ring of continuous functions on Rm×n and assume that for each f∈ℛ the sequence of compositions {f(∇uk)(1+|∇uk|p) dx}k∈N embedded into the space of measures on Ω¯ converges weakly * to some measure μf. We discuss the possibility to modify the sequence {uk}k∈N in such a way that the new sequence {wk}k∈N is still bounded in W1,p(Ω, Rm), converges weakly to u, each sequence of measures {f(∇wk)(1+|∇wk|p) dx}k∈N also converges weakly * to μf, where f∈ℛ, but additionally the new sequence satisfies the condition “wk=u” on F. Our results are applied to the minimization problems in the Calculus of Variations.
Keywords: sequences of gradients, DiPerna–Majda measures, concentrations, oscillations
