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R n x = Fx n (Rn X X R m by the formula C). 3) We have gr() = gr(F) n (X X Rn X C) E T ® B(Rn) ® B(Rm) . 4) x = cl{(~Pk(x) , gk(x)) : kEN} for every x E B. Consider the set M = {(x,~) EX x Rn: x E B,~ E cl{IPk(x): kEN}}. 5) we conclude easy that H- 1 (C) c M . To prove that M c H- 1 (C) , consider any fixed (x,O EM. 5), there exists a subsequence (IPu(k)) such that~= lim IPu(k)(x) . Since Cis a compact set and gk(x) E C, we may assume (passing to a further subsequence if necessary) that (gu(k)(x)) converges to some 1J E Rm.

1. 4 . As we pointed out earlier, the domain of any operator of the class M v coincides with the whole space V . 1. 1 Let B E Mv and D(B) :) C0 (Q) . Then there exists a unique multivalued function a associated to a. E Mq such that B c A, where A Proof . Let us define a subset E of U(Q)n Xu' (Q)n E Mv is the operator by E = {(V'v,g) E L(Q)P x U'(Qt : v. E V,g E Bu}. e. on Q. e. 18) Let us introduce the set dec(E) being the smallest decomposable set containing E. e. on Q;. 18), hold true with E replaced by dec(E).

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G-Convergence and Homogenization of Nonlinear Partial Differential Operators by Alexander Pankov

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Categories: Differential Equations