Semi-strong convergence of sequences satisfying variational inequality
Résumé
In this paper we study the properties of any sequence (un)n ≥ 1 weakly converging to a nonnegative function u in W1,p0(Ω), p>1, and satisfying a variational inequality of type -div(an(.,∇ un))≥ fn, where (an)n ≥ 1 is a suitable sequence of monotone operators and (fn)n ≥ 1 is any strongly convergent sequence in the dual space W-1,p'(Ω). We prove that the sequence (un - (1-ε)u)- strongly converges to 0 in W1,p0(Ω) for any ε>0. We show by a counter-example that the result does not hold true if ε=0. A remarkable corollary of these strong ε-convergences is that the sequence(u n)n ≥ 1 satisfies, up to a subsequence, a kind of semi-strong convergence: (un)n ≥ 1 can be bounded from below by a sequence which converges to the same limit u but strongly in W1,p0(Ω). We also give an example of a nonnegative weakly convergent sequence which does not satisfy this semi-strong convergence property and hence cannot satisfy any variational inequality of the previous type. Finally, in the linear case of a sequence of highly oscillating matrices, we improve the strong $\varepsilon$-convergences by replacing the arbitrary small constant ε>0 by a sequence(ε n)n ≥ 1 converging to 0.
Domaines
Analyse fonctionnelle [math.FA]
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