Functionals on Bv Space With Carath\'{e}odory Integrands Using Convex Duality

Wunderli, Thomas

doi:10.12732/ijpam.v112i1.13

Cited by 1 publication

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“…for a.e. x ∈ Ω and for each p ∈ ℝ N : As in the proof of Lemma 2 in [16] and Theorem 1 above, we have for a.e x, all q ∈ ℝ N ,…”

Section: Resultsmentioning

confidence: 97%

“…We finally remark, as noted in [1], that Theorems 1 and 4 of this paper may be extended to vector-valued functions uðxÞ = ðu 1 ðxÞ, ⋯, u M ðxÞÞ where Du is an M × N matrix with Du i ∈ BVðΩÞ for each i and Ð Ω gðx, DuÞ is defined by writing Du as a vector of length NM with g : Ω × ℝ NM ⟶ ℝ and g depending on ðx, jDujÞ for the case of Theorem 1. We may also consider integrands Ð Ω gðx, u, DuÞ with appropriate assumptions on gðx, z, pÞ, such as Lipschitz continuity in z, using similar methods as presented here and in [1,15,16].…”

Section: Resultsmentioning

confidence: 99%

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Lower Semicontinuity in $L^{1}$ of a Class of Functionals Defined on $B$

Wunderli

2021

Abstract and Applied Analysis

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We prove lower semicontinuity in L 1 Ω for a class of functionals G : B V Ω ⟶ ℝ of the form G u = ∫ Ω g x , ∇ u d x + ∫ Ω ψ x d D s u where g : Ω × ℝ N ⟶ ℝ , Ω ⊂ ℝ N is open and bounded, g · , p ∈ L 1 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 ∈ B V Ω ; the gradient measure D u = ∇ u d x + d D s u x is decomposed into mutually singular measures ∇ u d x and d D s u x . As an example, we use this to prove that ∫ Ω ψ x α 2 x + ∇ u 2 d x + ∫ Ω ψ x d D s u is lower semicontinuous in L 1 Ω for any bounded continuous ψ and any α ∈ L 1 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 − u 0 L 1 for the given u 0 ∈ L 1 Ω , due to the compactness of B V Ω in L 1 Ω .

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“…for a.e. x ∈ Ω and for each p ∈ ℝ N : As in the proof of Lemma 2 in [16] and Theorem 1 above, we have for a.e x, all q ∈ ℝ N ,…”

Section: Resultsmentioning

confidence: 97%