English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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 .

abstract and applied analysis

Lower Semicontinuity in <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi>L</mi> </mrow> <mrow> <mn>1</mn> </mrow> </msup> </math> of a Class of Functionals Defined on <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>B</mi> <mi>V</mi> </math> with Carathéodory Integrands

Lower Semicontinuity in L 1 of a Class of Functionals Defined on B V with Carathéodory Integrands

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