On the definition and the lower semicontinuity of certain quasiconvex integrals

摘要: Let us consider vector-valued functions u:Ω→ℝN, defined in an open bounded set Ω⊂ℝn. f(x, ξ) be a continuous function Ω×ℝnN, quasiconvex with respect on ξ, that satisfies, for some p ≦ q, the growth conditions c1|ξ|p c2(1 + |ξ|q). The integral I(u)=∫Ωf(x,Du(x))dx is well if u∈H1,q(Ω;ℝN). We extend I(u) to u∈H1,p(Ω;ℝN), and we study its lower semicontinuity weak topology of H1,p(Ω;ℝN), order obtain existence minima.