\(H^{1/2}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation

DOI10.1007/s10240-004-0019-5zbMath1051.49030OpenAlexW2066518895MaRDI QIDQ1880957

Petru Mironescu, Jean Bourgain, Haim Brezis

Publication date: 27 September 2004

Published in: Publications Mathématiques (Search for Journal in Brave)

Full work available at URL: http://www.numdam.org/item?id=PMIHES_2004__99__1_0

zbMATH Keywords

lifting minimization problem Ginzburg-Landau equation topological degree minimal connection

Mathematics Subject Classification ID

Asymptotic behavior of solutions to PDEs (35B40) Variational problems in a geometric measure-theoretic setting (49Q20) Nonlinear elliptic equations (35J60) Existence of solutions for minimax problems (49J35) Degree theory for nonlinear operators (47H11) Statistical mechanics of superconductors (82D55) Statistical mechanics of superfluids (82D50) Variational methods for second-order elliptic equations (35J20) Infinite-dimensional manifolds (46T05)

Related Items (76)

Integrability of the Brouwer degree for irregular arguments ⋮ New estimates for the Laplacian, the div-curl, and related Hodge systems ⋮ Minimizers of the \(W^{1, 1}\)-energy of \(\mathbb{S}^1\)-valued maps with prescribed singularities. Do they exist? ⋮ Sobolev bundles with Abelian structure groups ⋮ A remark on precomposition on \(\mathrm H^{1/2}(S^1)\) and \(\varepsilon\)-identifiability of disks in tomography ⋮ Partial regularity for \(\omega\)-minimizers of quasiconvex functionals ⋮ Energy with weight for \(S^2\)-valued maps with prescribed singularities ⋮ Two remarks on liftings of maps with values into \(S^1\). ⋮ Unnamed Item ⋮ The homological singularities of maps in trace spaces between manifolds ⋮ New monotonicity formulae for semi-linear elliptic and parabolic systems ⋮ An optimal Sobolev embedding for \(L^1\) ⋮ Nearly parallel vortex filaments in the 3D Ginzburg-Landau equations ⋮ Limiting Sobolev and Hardy inequalities on stratified homogeneous groups ⋮ Phases of unimodular complex valued maps: optimal estimates, the factorization method, and the sum-intersection property of Sobolev spaces ⋮ Two-dimensional unit-length vector fields of vanishing divergence ⋮ Norming and dense sets of extreme points of the unit ball in spaces of bounded Lipschitz functions ⋮ Vortex density models for superconductivity and superfluidity ⋮ Some of my favorite open problems ⋮ A brief history of the Jacobian ⋮ An embedding theorem for BV-functions ⋮ A variant of Poincaré's inequality. ⋮ Remarks on the Monge-Kantorovich problem in the discrete setting ⋮ The Jacobian determinant revisited ⋮ Endpoint \(L^1\) estimates for Hodge systems ⋮ Gradient vector fields with values into \(S^1\) ⋮ On an open problem for Jacobians raised by Bourgain, Brezis and mironescu. ⋮ On the distributional Jacobian of maps from \(\mathbb{S}^N\) into \(\mathbb{S}^N\) in fractional Sobolev and Hölder spaces ⋮ Three dimensional vortex approximation construction and \({\epsilon}\)-level estimates for the Ginzburg-Landau functional ⋮ Study of a \(3D\)-Ginzburg-Landau functional with a discontinuous pinning term ⋮ \(\mathbb S^1\)-valued Sobolev maps ⋮ On some refinements of the embedding of critical Sobolev spaces into BMO ⋮ Uniform boundedness principles for Sobolev maps into manifolds ⋮ A simple proof of an inequality of Bourgain, Brezis and Mironescu. ⋮ Lifting of BV functions with values in \(S^1\) ⋮ An estimate in the spirit of Poincaré's inequality ⋮ Partial regularity for symmetric quasiconvex functionals on BD ⋮ Strong approximation of fractional Sobolev maps ⋮ A new obstruction to the extension problem for Sobolev maps between manifolds ⋮ Estimates for the topological degree and related topics ⋮ Partial regularity for BV minimizers ⋮ \(W^{s,\frac{n}{s}}\)-maps with positive distributional Jacobians ⋮ Lifting in compact covering spaces for fractional Sobolev mappings ⋮ Lifting in Besov spaces ⋮ Inequalities related to liftings and applications ⋮ Continuity and canceling operators of order \(n\) on \(\mathbb{R}^n\) ⋮ Estimates for \(L^{1}\)-vector fields under higher-order differential conditions ⋮ Mean field limits for Ginzburg-Landau vortices ⋮ An \(L^1\)-type estimate for Riesz potentials ⋮ Topological singularities in \(W^{S,P}(S^N,S^{1})\) ⋮ Limiting Bourgain-Brezis estimates for systems of linear differential equations: theme and variations ⋮ Symmetry in nonlinear PDEs: some open problems ⋮ The space \(\operatorname{BV}(S^2,S^1)\): minimal connection and optimal lifting ⋮ Some inequalities related to Sobolev norms ⋮ A variational problem with lack of compactness for \(H^{1/2}(S^{1};S^{1})\) maps of prescribed degree ⋮ The relaxed energy for \(S^2\)-valued maps and measurable weights ⋮ On the distributions of the form \(\sum_i(\delta_{p_i}-\delta_{n_i})\) ⋮ Optimal lifting for BV \(( S^{1}, S^{1})\) ⋮ Function spaces between BMO and critical Sobolev spaces ⋮ On the distributions of the form \(\sum_i(\delta_{p_i}-\delta_{n_i})\) ⋮ Convergence of Ginzburg-Landau functionals in three-dimensional superconductivity ⋮ Decomposition of \(\mathbb S^1\)-valued maps in Sobolev spaces ⋮ New estimates for elliptic equations and Hodge type systems ⋮ Density in \(W^{s, p}(\operatorname{\Omega}; N)\) ⋮ Trace theory for Sobolev mappings into a manifold ⋮ Minimizing \(1/2\)-harmonic maps into spheres ⋮ Prescribing the Jacobian in critical spaces ⋮ Strong density results in trace spaces of maps between manifolds ⋮ THE GINZBURG–LANDAU FUNCTIONAL WITH A DISCONTINUOUS AND RAPIDLY OSCILLATING PINNING TERM. PART I: THE ZERO DEGREE CASE ⋮ Where Sobolev interacts with Gagliardo-Nirenberg ⋮ A product-estimate for Ginzburg-Landau and corollaries ⋮ Embeddings for \(\mathbb{A}\)-weakly differentiable functions on domains ⋮ On sequences of maps into a manifold with equibounded \(W^{1/2}\)-energies ⋮ Density results for the \(W^{1/2}\) energy of maps into a manifold ⋮ Dimension estimates for vectorial measures with restricted spectrum ⋮ Bourgain-Brezis inequalities on symmetric spaces of non-compact type

Cites Work

This page was built for publication: \(H^{1/2}\) maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation