Relaxation and integral representation for functionals of linear growth on metric measure spaces

DOI10.1515/agms-2016-0013zbMath1354.49027arXiv1401.5717OpenAlexW2492393789MaRDI QIDQ2520426

Panu Lahti, Heikki Hakkarainen, Pekka Lehtelä, Juha Kinnunen

Publication date: 13 December 2016

Published in: Analysis and Geometry in Metric Spaces (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1401.5717

zbMATH Keywords

integral representationrelaxationcalculus of variationsfunctions of bounded variationfunctionalslinear growthmetric measure spaces

Mathematics Subject Classification ID

Variational problems in a geometric measure-theoretic setting (49Q20) Methods involving semicontinuity and convergence; relaxation (49J45) Absolutely continuous real functions of several variables, functions of bounded variation (26B30) Analysis on metric spaces (30L99)

Related Items (22)

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces ⋮ Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality ⋮ Lower semicontinuity for an integral functional in BV ⋮ An analog of the Neumann problem for the 1-Laplace equation in the metric setting: existence, boundary regularity, and stability ⋮ Strong \textit{BV}-extension and \(W^{1,1}\)-extension domains ⋮ \(\Gamma\)-convergence of nonconvex integrals in Cheeger-Sobolev spaces and homogenization ⋮ Neumann \(p\)-Laplacian problems with a reaction term on metric spaces ⋮ Relaxation of nonconvex unbounded integrals with general growth conditions in Cheeger-Sobolev spaces ⋮ The Neumann and Dirichlet problems for the total variation flow in metric measure spaces ⋮ A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces ⋮ Capacitary density and removable sets for Newton-Sobolev functions in metric spaces ⋮ Removable sets for Newtonian Sobolev spaces and a characterization of \(p\)-path almost open sets ⋮ BV and Sobolev homeomorphisms between metric measure spaces and the plane ⋮ Strong approximation of sets of finite perimeter in metric spaces ⋮ A sharp Leibniz rule for BV functions in metric spaces ⋮ Trace theorems for functions of bounded variation in metric spaces ⋮ Asymptotic behavior of $BV$ functions and sets of finite perimeter in metric measure spaces ⋮ Perimeter as relaxed Minkowski content in metric measure spaces ⋮ Traces of Newtonian-Sobolev, Hajlasz-Sobolev, and BV functions on metric spaces ⋮ Notions of Dirichlet problem for functions of least gradient in metric measure spaces ⋮ Discrete convolutions of \(\text{BV}\) functions in quasiopen sets in metric spaces ⋮ Stability and continuity of functions of least gradient

Cites Work

• Pointwise properties of functions of bounded variation in metric spaces
• Extensions and traces of functions of bounded variation on metric spaces
• Regularity of minimizers of the area functional in metric spaces
• Fine properties and a notion of quasicontinuity for BV functions on metric spaces
• Nonlinear potential theory on metric spaces
• A characterization of Newtonian functions with zero boundary values
• Functions of bounded variation on ``good metric spaces
• Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces
• Integral representation of convex functions on a space of measures
• Fine properties of sets of finite perimeter in doubling metric measure spaces
• Definitions of Sobolev classes on metric spaces
• Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
• Stability and continuity of functions of least gradient
• Geometric Properties of Planar BV -Extension Domains
• Sobolev Extensions of Hölder Continuous and Characteristic Functions on Metric Spaces
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