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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| المؤلف الرئيسي: | |
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| التنسيق: | article |
| منشور في: |
2021
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| الوصول للمادة أونلاين: | http://hdl.handle.net/11073/24057 |
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| _version_ | 1864513442409349120 |
|---|---|
| author | Wunderli, Thomas |
| author_facet | Wunderli, Thomas |
| author_role | author |
| dc.creator.none.fl_str_mv | Wunderli, Thomas |
| dc.date.none.fl_str_mv | 2021 2022-06-23T07:50:38Z 2022-06-23T07:50:38Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | T. Wunderli, "Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands", Abstract and Applied Analysis, vol. 2021, Article ID 6709303, 6 pages, 2021. https://doi.org/10.1155/2021/6709303 1085-3375 http://hdl.handle.net/11073/24057 10.1155/2021/6709303 |
| dc.language.none.fl_str_mv | en_US |
| dc.publisher.none.fl_str_mv | Hindawi Limited |
| dc.relation.none.fl_str_mv | https://doi.org/10.1155/2021/6709303 |
| dc.title.none.fl_str_mv | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| dc.type.none.fl_str_mv | Peer-Reviewed Published version info:eu-repo/semantics/publishedVersion info:eu-repo/semantics/article |
| description | 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). |
| format | article |
| id | aus_d2335cb6902cea452d9da7fde410044e |
| identifier_str_mv | T. Wunderli, "Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands", Abstract and Applied Analysis, vol. 2021, Article ID 6709303, 6 pages, 2021. https://doi.org/10.1155/2021/6709303 1085-3375 10.1155/2021/6709303 |
| language_invalid_str_mv | en_US |
| network_acronym_str | aus |
| network_name_str | aus |
| oai_identifier_str | oai:repository.aus.edu:11073/24057 |
| publishDate | 2021 |
| publisher.none.fl_str_mv | Hindawi Limited |
| repository.mail.fl_str_mv | |
| repository.name.fl_str_mv | |
| repository_id_str | |
| spelling | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory IntegrandsWunderli, ThomasWe 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).Hindawi Limited2022-06-23T07:50:38Z2022-06-23T07:50:38Z2021Peer-ReviewedPublished versioninfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articleapplication/pdfT. Wunderli, "Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands", Abstract and Applied Analysis, vol. 2021, Article ID 6709303, 6 pages, 2021. https://doi.org/10.1155/2021/67093031085-3375http://hdl.handle.net/11073/2405710.1155/2021/6709303en_UShttps://doi.org/10.1155/2021/6709303oai:repository.aus.edu:11073/240572024-08-22T12:01:56Z |
| spellingShingle | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands Wunderli, Thomas |
| status_str | publishedVersion |
| title | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| title_full | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| title_fullStr | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| title_full_unstemmed | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| title_short | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| title_sort | Lower Semicontinuity in L¹ of a Class of Functionals Defined on BV with Caratheodory Integrands |
| url | http://hdl.handle.net/11073/24057 |