Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 3 (2014), 627-671.

The 1-harmonic flow with values in a hyperoctant of the $N$-sphere

Lorenzo Giacomelli, Jose Mazón, and Salvador Moll

Full-text: Open access

Abstract

We prove the existence of solutions to the 1-harmonic flow — that is, the formal gradient flow of the total variation of a vector field with respect to the L2-distance — from a domain of m into a hyperoctant of the N-dimensional unit sphere, S+N1, under homogeneous Neumann boundary conditions. In particular, we characterize the lower-order term appearing in the Euler–Lagrange formulation in terms of the “geodesic representative” of a BV-director field on its jump set. Such characterization relies on a lower semicontinuity argument which leads to a nontrivial and nonconvex minimization problem: to find a shortest path between two points on S+N1 with respect to a metric which penalizes the closeness to their geodesic midpoint.

Article information

Source
Anal. PDE, Volume 7, Number 3 (2014), 627-671.

Dates
Received: 18 April 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731523

Digital Object Identifier
doi:10.2140/apde.2014.7.627

Mathematical Reviews number (MathSciNet)
MR3227428

Zentralblatt MATH identifier
1295.35260

Subjects
Primary: 35K55: Nonlinear parabolic equations 49Q20: Variational problems in a geometric measure-theoretic setting 53C22: Geodesics [See also 58E10] 35K67: Singular parabolic equations 35K92: Quasilinear parabolic equations with p-Laplacian
Secondary: 49J45: Methods involving semicontinuity and convergence; relaxation 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 58E20: Harmonic maps [See also 53C43], etc. 68U10: Image processing

Keywords
harmonic flows total variation flow nonlinear parabolic systems lower semicontinuity and relaxation nonconvex variational problems geodesics Riemannian manifolds with boundary image processing

Citation

Giacomelli, Lorenzo; Mazón, Jose; Moll, Salvador. The 1-harmonic flow with values in a hyperoctant of the $N$-sphere. Anal. PDE 7 (2014), no. 3, 627--671. doi:10.2140/apde.2014.7.627. https://projecteuclid.org/euclid.apde/1513731523


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