Variational problems for maps of bounded variation with values in \(S^ 1\)
type: Item
instance of: scholarly article
title: Variational problems for maps of bounded variation with values in \(S^ 1\)
zbMATH Open document ID: 0810.49040
DOI: 10.1007/BF02163266
author: Mariano Giaquinta
author: Giuseppe Modica
author: Jiří Souček
published in: Calculus of Variations and Partial Differential Equations
publication date: 1995-04-20T00:00:00Z
review text: The authors' main aim is to characterize the weak limits of smooth maps from an open subset \(\Omega\) of a Riemannian manifold into \(S^ 1\) (the unit circle). This is achieved in the class of Cartesian currents in \(\Omega\times S^ 1\), \(\text{cart}(\Omega\times S^ 1)\), introduced earlier by these authors [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 16, No. 3, 393-485 (1989; Zbl 0713.49014)]. In the present paper it is shown, among others, that each element of \(\text{cart}(\Omega\times S^ 1)\) is the weak limit of a sequence of smooth maps (from \(\Omega\) into \(S^ 1\)) with bounded variations. This result is then applied to prove the existence of minimizers of certain integral functionals in a subclass of \(\text{cart}(\Omega\times S^ 1)\). In this spirit, the dipole problem (minimizing the total variation under prescribed homological singularities) is solved.
Mathematics Subject Classification ID: 49Q20
zbMATH DE Number: 509971
zbMATH Keywords: variational problems on manifolds
zbMATH Keywords: Cartesian currents
zbMATH Keywords: existence of minimizers
zbMATH Keywords: integral functionals
zbMATH Keywords: dipole problem
reviewed by: Winfried Schirotzek
MaRDI profile type: Publication
cites work: Q3788768
cites work: A General Chain Rule for Distributional Derivatives
cites work: Funzioni BV e tracce
cites work: Harmonic maps with defects
cites work: On a partial differential equation involving the Jacobian determinant
cites work: Integral representation on \(BV(\Omega )\) of \(\Gamma\)-limits of variational integrals
cites work: Su una teoria generale della misura \((r-1)\)-dimensionale in uno spazio ad \(r\) dimensioni
cites work: Nuovi teoremi relativi alle misure \((r - 1)\)-dimensionali in uno spazio ad \(r\) dimensioni
cites work: Q5565773
cites work: Variational problems on vector bundles
cites work: Q4196944
cites work: Cartesian currents, weak diffeomorphisms and existence theorems in nonlinear elasticity
cites work: Q3198418
cites work: The Dirichlet energy of mappings with values into the sphere
cites work: The Dirichlet integral for mappings between manifolds: Cartesian currents and homology
cites work: Q5531732
cites work: Q3992482
cites work: Q3336268
full work available at URL: https://doi.org/10.1007/bf02163266
OpenAlex ID: W2057464224
Recommended article: A Variational Problem Pertaining to Biharmonic Maps
Recommended article: A variational problem with lack of compactness for \(H^{1/2}(S^{1};S^{1})\) maps of prescribed degree
Recommended article: Q4647922
Recommended article: Variational integrals on mappings of bounded variation and their lower semicontinuity
Recommended article: On mappings of bounded variation
Recommended article: A variational approach to \(S^1\)-harmonic maps and applications
Recommended article: VARIATIONAL PROBLEMS OF CERTAIN FUNCTIONALS
Recommended article: On maps of bounded \(p\)-variation with \(p>1\)
Recommended article: Maps of bounded variation with values into a manifold: total variation and relaxed energy
Recommended article: Q3159182
description: scientific article; zbMATH DE number 509971