New lower semicontinuity results for polyconvex integrals

Summary

We study integral functionals of the formF(u, Ω)=∫Ω f(▽u)dx, defined foru ∈ C¹(Ω;R ^k), Ω⊑R ⁿ. The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) ≥c₀¦ℳ(A)¦ for a suitable constant c₀ > 0, where ℳ(A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(Ω;R ^k) with respect to the strong topology ofL ¹(Ω;R ^k). Then we consider the relaxed functional ℱ, defined as the greatest lower semicontinuous functional onL ¹(Ω;R ^k) which is less than or equal toF on C¹(Ω;R ^k). For everyu ∈ BV(Ω;R ^k) we prove that ℱ (u,Ω) ≥ ∫Ω f(▽u)dx+c₀¦D^su¦(Ω), whereDu=▽u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that ℱ (u,Ω)=∫Ω f(▽u)dx for everyu ∈ W^1,p(Ω;R ^k), withp ≥ min{n,k}. We prove also that the functional ℱ (u, Ω) can not be represented by an inte- gral for an arbitrary functionu ∈ BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set functionΩ → ℱ (u, Ω) is not subadditive whenu ∈ BV_loc(R ⁿ;R ^k), even ifu ∈ W ^1,p_loc (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu ∈ BV(Ω;R ^k) such that ℱ (u, Ω)=∫Ω f(▽u)dx, particularly in the model casef(A)=¦ℳ(A)¦.