Relaxation and optimization for linear-growth convex integral functionals under PDE constraints.
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| Title: | Relaxation and optimization for linear-growth convex integral functionals under PDE constraints. |
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| Authors: | Arroyo-Rabasa, Adolfo1 adolfo.arroyo.rabasa@hcm.uni-bonn.de |
| Source: | Journal of Functional Analysis. Oct2017, Vol. 273 Issue 7, p2388-2427. 40p. |
| Subjects: | Integral functions, Saddlepoint approximations, Integrable functions, Partial differential equations, Radon measures |
| Abstract: | We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the form F [ u ] = ∫ Ω f ( x , u ( x ) ) d x , u : Ω ⊂ R d → R N , where f : Ω × R N → R is a convex integrand and u is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures M ( Ω ; R N ) , and the introduction of a set-valued pairing on M ( Ω ; R N ) × L ∞ ( Ω ; R N ) . By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD and divergence-free spaces. [ABSTRACT FROM AUTHOR] |
| Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
| FullText | Text: Availability: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Relaxation and optimization for linear-growth convex integral functionals under PDE constraints. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Arroyo-Rabasa%2C+Adolfo%22">Arroyo-Rabasa, Adolfo</searchLink><relatesTo>1</relatesTo><i> adolfo.arroyo.rabasa@hcm.uni-bonn.de</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Journal+of+Functional+Analysis%22">Journal of Functional Analysis</searchLink>. Oct2017, Vol. 273 Issue 7, p2388-2427. 40p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Integral+functions%22">Integral functions</searchLink><br /><searchLink fieldCode="DE" term="%22Saddlepoint+approximations%22">Saddlepoint approximations</searchLink><br /><searchLink fieldCode="DE" term="%22Integrable+functions%22">Integrable functions</searchLink><br /><searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Radon+measures%22">Radon measures</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We give necessary and sufficient conditions for the minimality of generalized minimizers of linear-growth integral functionals of the form F [ u ] = ∫ Ω f ( x , u ( x ) ) d x , u : Ω ⊂ R d → R N , where f : Ω × R N → R is a convex integrand and u is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures M ( Ω ; R N ) , and the introduction of a set-valued pairing on M ( Ω ; R N ) × L ∞ ( Ω ; R N ) . By these means we are able to show an intrinsic relation between minimizers of the relaxed problem and maximizers of its dual formulation also known as the saddle-point conditions. In particular, our results can be applied to relaxation and minimization problems in BV, BD and divergence-free spaces. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1016/j.jfa.2017.06.012 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 40 StartPage: 2388 Subjects: – SubjectFull: Integral functions Type: general – SubjectFull: Saddlepoint approximations Type: general – SubjectFull: Integrable functions Type: general – SubjectFull: Partial differential equations Type: general – SubjectFull: Radon measures Type: general Titles: – TitleFull: Relaxation and optimization for linear-growth convex integral functionals under PDE constraints. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Arroyo-Rabasa, Adolfo IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 10 Text: Oct2017 Type: published Y: 2017 Identifiers: – Type: issn-print Value: 00221236 Numbering: – Type: volume Value: 273 – Type: issue Value: 7 Titles: – TitleFull: Journal of Functional Analysis Type: main |
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