Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

AbstractThis article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is defined via relaxation, and it defines a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.

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Regularity of minimizers of the area functional in metric spaces

Advances in Calculus of Variations ◽

10.1515/acv-2013-0022 ◽

2015 ◽

Vol 8 (1) ◽

Cited By ~ 4

Author(s):

Heikki Hakkarainen ◽

Juha Kinnunen ◽

Panu Lahti

Keyword(s):

Linear Growth ◽

Metric Spaces ◽

Metric Measure Spaces ◽

Local Boundedness ◽

Regularity Of Minimizers ◽

Jump Discontinuities ◽

Area Functional ◽

Measure Spaces

AbstractIn this article we study minimizers of functionals of linear growth in metric measure spaces. We introduce the generalized problem in this setting, and prove existence and local boundedness of the minimizers. We give counterexamples to show that in general, minimizers are not continuous and can have jump discontinuities inside the domain.

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On Trudinger-type Inequalities in Orlicz-Morrey Spaces of an Integral Form

Canadian Mathematical Bulletin ◽

10.4153/s0008439520000223 ◽

2020 ◽

pp. 1-16

Author(s):

Ritva Hurri-Syrjänen ◽

Takao Ohno ◽

Tetsu Shimomura

Keyword(s):

Integral Form ◽

Morrey Spaces ◽

Metric Measure Spaces ◽

Riesz Potentials ◽

Euclidean Case ◽

Measure Spaces

Abstract We give Trudinger-type inequalities for Riesz potentials of functions in Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our results are new even for the doubling metric measure setting. In particular, our results improve and extend the previous results in Morrey spaces of an integral form in the Euclidean case.

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On the relaxation of variational integrals in metric Sobolev spaces

Advances in Calculus of Variations ◽

10.1515/acv-2013-0207 ◽

2015 ◽

Vol 8 (1) ◽

Cited By ~ 1

Author(s):

Omar Anza Hafsa ◽

Jean-Philippe Mandallena

Keyword(s):

Integral Representation ◽

Calculus Of Variations ◽

Sobolev Spaces ◽

General Framework ◽

Metric Measure Spaces ◽

Representation Theorems ◽

Differentiable Structure ◽

Convex Case ◽

Measure Spaces ◽

Variational Integrals

AbstractWe give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for “relaxed” variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger–Keith's differentiable structure.

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