It is proved that if u ∈ BHp(Ω;ℝd), with p > 1, if {un} is bounded in BHp(Ω; ℝd), |Ds2un|(Ω) → 0, and if un → u in W1, 1(Ω; ℝd), then ∫ Ω f(x, u(x), ∇u(x), ∇2u(x)), dx ≤ lim infn→+∞ ∫Ω f(x, un (x), ∇un(x), ∇2un(x)) dx provided f(x, u, ξ, ·) is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BHP are provided.
On lower semicontinuity in BHp and 2-quasiconvexification
Paroni, Roberto
2003-01-01
Abstract
It is proved that if u ∈ BHp(Ω;ℝd), with p > 1, if {un} is bounded in BHp(Ω; ℝd), |Ds2un|(Ω) → 0, and if un → u in W1, 1(Ω; ℝd), then ∫ Ω f(x, u(x), ∇u(x), ∇2u(x)), dx ≤ lim infn→+∞ ∫Ω f(x, un (x), ∇un(x), ∇2un(x)) dx provided f(x, u, ξ, ·) is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BHP are provided.File in questo prodotto:
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