Analysis & PDE
- Anal. PDE
- Volume 7, Number 3 (2014), 627-671.
The 1-harmonic flow with values in a hyperoctant of the $N$-sphere
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 -distance — from a domain of into a hyperoctant of the -dimensional unit sphere, , 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 with respect to a metric which penalizes the closeness to their geodesic midpoint.
Anal. PDE, Volume 7, Number 3 (2014), 627-671.
Received: 18 April 2013
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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
harmonic flows total variation flow nonlinear parabolic systems lower semicontinuity and relaxation nonconvex variational problems geodesics Riemannian manifolds with boundary image processing
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