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Lower Semicontinuity in Lยน of a Class of Functionals Defined on BV with Caratheodory Integrands
Wunderli, Thomas
Wunderli, Thomas
Date
2021
Authors
Wunderli, Thomas
Advisor
Type
Article
Peer-Reviewed
Published version
Peer-Reviewed
Published version
Degree
Description
Abstract
We prove lower semicontinuity in ๐ฟยน(ฮฉ) for a class of functionals ๐ข :๐ต๐(ฮฉ) โโ of the form ๐ข(๐ข)=โซฮฉ๐(๐ฅ, ๐ป๐ข)๐๐ฅ + โซฮฉ๐(๐ฅ)๐|Dหข๐ข| where ๐ :ฮฉโจโแดบโโ, ฮฉโโแดบ is open and bounded, ๐(.,๐) โ ๐ฟยน(ฮฉ) for each ๐ satisfies the linear growth condition lim|๐โโ ๐(๐ฅ,๐)/|๐| = ๐(๐ฅ) โ ๐ถ(ฮฉ) โฉ ๐ฟโ (ฮฉ) and is convex in ๐ depending only on |๐| for a.e. ๐ฅ. Here, we recall for ๐ข โ ๐ต๐(ฮฉ); the gradient measure ๐ท๐ข = ๐ป๐ข ๐๐ฅ + ๐(Dหข๐ข)(๐ฅ) is decomposed into mutually singular measures ๐ป๐ข ๐๐ฅ and ๐(Dหข๐ข)(๐ฅ). As an example, we use this to prove that โซฮฉ๐(๐ฅ) โ[๐ผยฒ(๐ฅ) + | ๐ป๐ข |ยฒ ๐๐ฅ + โซฮฉ๐(๐ฅ)๐|Dหข๐ข|] is lower semicontinuous in ๐ฟยน(ฮฉ) for any bounded continuous ๐ and any ๐ผ โ ๐ฟยน(ฮฉ). Under minor addtional assumptions on ๐, we then have the existence of minimizers of functionals to variational problems of the form ๐ข(๐ข) + ||๐ข - ๐ขโ||๐ฟยน for the given ๐ขโ โ ๐ฟยน(ฮฉ) due to the compactness of ๐ต๐(ฮฉ) in ๐ฟยน(ฮฉ).