New Results on Γ-limits of Integral Functionals
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چکیده
For ψ ∈ W 1,p(Ω;Rm) and g ∈ W−1,p(Ω;Rd), 1 < p < +∞, we consider a sequence of integral functionals F k : W 1,p(Ω;Rm)× Lp(Ω;Rd×n)→ [0,+∞] of the form F k (u, v) = ∫ Ω fk(x,∇u, v) dx if u− ψ ∈W 1,p 0 (Ω;Rm) and div v = g, +∞ otherwise, where the integrands fk satisfy growth conditions of order p, uniformly in k. We prove a Γ-compactness result for F k with respect to the weak topology of W 1,p(Ω;Rm) × Lp(Ω;Rd×n) and we show that under suitable assumptions the integrand of the Γ-limit is continuously differentiable. We also provide a result concerning the convergence of momenta for minimizers of F k . Résumé. Pour tout ψ ∈ W 1,p(Ω;Rm) et g ∈ W−1,p(Ω;Rd), 1 < p < +∞, nous considérons une suite de fonctionnelles intégrale F k : W 1,p(Ω;Rm)× Lp(Ω;Rd×n)→ [0,+∞] définies par F k (u, v) = ∫ Ω fk(x,∇u, v) dx si u− ψ ∈W 1,p 0 (Ω;Rm) et div v = g, +∞ sinon, où les intégrandes fk satisfont des conditions de croissance d’ordre p, uniformément en k. Nous démontrons un résultat de Γ-compacité pour F k par rapport à la topologie faible sur W 1,p(Ω;Rm)× Lp(Ω;Rd×n) et nous prouvons que sous des conditions appropriées, l’intégrande de la Γ-limite est continûment différentiable. Nous montrons également un résultat de convergence des moments pour les minima de F k .
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تاریخ انتشار 2013