Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands
We prove lower semicontinuity in ¹(Ω) for a class of functionals :(Ω) →ℝ of the form ()=∫Ω(, ) + ∫Ω()|Dˢ| where :Ω⨉ℝᴺ→ℝ, Ω⊂ℝᴺ is open and bounded, (.,) ∊ ¹(Ω) for each satisfies the linear growth condition lim|→∞ (,)/|| = () ∊ (Ω) ∩ ∞ (Ω) and is convex in depending only on || for a.e. . Here, we rec...
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| Format: | article |
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2021
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| Online Access: | http://hdl.handle.net/11073/24057 |
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| Summary: | We prove lower semicontinuity in ¹(Ω) for a class of functionals :(Ω) →ℝ of the form ()=∫Ω(, ) + ∫Ω()|Dˢ| where :Ω⨉ℝᴺ→ℝ, Ω⊂ℝᴺ is open and bounded, (.,) ∊ ¹(Ω) for each satisfies the linear growth condition lim|→∞ (,)/|| = () ∊ (Ω) ∩ ∞ (Ω) and is convex in depending only on || for a.e. . Here, we recall for ∊ (Ω); the gradient measure = + (Dˢ)() is decomposed into mutually singular measures and (Dˢ)(). As an example, we use this to prove that ∫Ω() √[²() + | |² + ∫Ω()|Dˢ|] is lower semicontinuous in ¹(Ω) for any bounded continuous and any ∊ ¹(Ω). Under minor addtional assumptions on , we then have the existence of minimizers of functionals to variational problems of the form () + || - ₀||¹ for the given ₀ ∊ ¹(Ω) due to the compactness of (Ω) in ¹(Ω). |
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