Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands

Abstract

We prove lower semicontinuity in 𝐿¹(Ω) for a class of functionals 𝒢 :𝐵𝑉(Ω) →ℝ of the form 𝒢(𝑢)=∫Ω𝑔(𝑥, 𝛻𝑢)𝑑𝑥 + ∫Ω𝜓(𝑥)𝑑|Dˢ𝑢| where 𝑔 :Ω⨉ℝᴺ→ℝ, Ω⊂ℝᴺ is open and bounded, 𝑔(.,𝑝) ∊ 𝐿¹(Ω) for each 𝑝 satisfies the linear growth condition lim|𝑝→∞ 𝑔(𝑥,𝑝)/|𝑝| = 𝜓(𝑥) ∊ 𝐶(Ω) ∩ 𝐿∞ (Ω) and is convex in 𝑝 depending only on |𝑝| for a.e. 𝑥. Here, we recall for 𝑢 ∊ 𝐵𝑉(Ω); the gradient measure 𝐷𝑢 = 𝛻𝑢 𝑑𝑥 + 𝑑(Dˢ𝑢)(𝑥) is decomposed into mutually singular measures 𝛻𝑢 𝑑𝑥 and 𝑑(Dˢ𝑢)(𝑥). As an example, we use this to prove that ∫Ω𝜓(𝑥) √[𝛼²(𝑥) + | 𝛻𝑢 |² 𝑑𝑥 + ∫Ω𝜓(𝑥)𝑑|Dˢ𝑢|] is lower semicontinuous in 𝐿¹(Ω) for any bounded continuous 𝜓 and any 𝛼 ∊ 𝐿¹(Ω). Under minor addtional assumptions on 𝑔, we then have the existence of minimizers of functionals to variational problems of the form 𝒢(𝑢) + ||𝑢 - 𝑢₀||𝐿¹ for the given 𝑢₀ ∊ 𝐿¹(Ω) due to the compactness of 𝐵𝑉(Ω) in 𝐿¹(Ω).