## Analysis & PDE

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

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

#### 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 $ℝm$ into a hyperoctant of the $N$-dimensional unit sphere, $S+N−1$, 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 $S+N−1$ 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
Accepted: 27 November 2013
First available in Project Euclid: 20 December 2017

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

#### 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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