. Proof, By Lemma 6.3.2, the sequence (E(u n )) n?N * is convergent

. Firstly, u), since u is the global minimizer of the functional E. By letting n go to infinity, we obtain E ? E(u

V. Let, M. , and ?. N-*,-u-n-v-?-m, Its existence is ensured by Lemma 6.3.3 M + 2 + u V ) be the closed ball of V centered at 0 of radius M + 2 + u V . Let L be the Lipschitz constant associated with K in (6.9) Using (6.9) and the fact that E ? (u) = 0, we have E ? (u n ) V ? Lu ? u n V and as (u n ) n?N * is bounded in V by Lemma 6.3.3, we deduce that (E ? (u n )) n?N * is also bounded in V . We can then extract a subsequence of (E ? (u n )) n?N * which weakly converges in V towards w ? V, Let us first prove that (E ?

. Then and . Span, is dense in V with assumption (A1), necessarily w = 0. Thus the sequence (E ? (u n )) n?N * weakly converges to 0 in V . As E is convex, we have the following inequality for all n

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