6533b829fe1ef96bd128ae38
RESEARCH PRODUCT
THE 1-HARMONIC FLOW WITH VALUES IN A HYPEROCTANT OF THE N-SPHERE
Salvador MollLorenzo GiacomelliJosé M. Mazónsubject
Unit spherenonconvex variational problemsriemannian manifolds with boundaryGeodesicn-sphereharmonic flows68U1053C2253C4435K9235K67Neumann boundary conditionpartial differential equations49J45MathematicsNumerical Analysisnonlinear parabolic systems; lower semicontinuity and relaxation; total variation flow; 1-harmonic flow; image processing; harmonic flows; partial differential equations; image processing.; geodesics; riemannian manifolds with boundary; nonconvex variational problemslower semicontinuity and relaxation58E20Applied MathematicsMathematical analysis49Q201-harmonic flowimage processingFlow (mathematics)35K55Metric (mathematics)total variation flowVector fieldnonlinear parabolic systemsBalanced flowAnalysisgeodesicsdescription
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 [math] -distance — from a domain of [math] into a hyperoctant of the [math] -dimensional unit sphere, [math] , 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 [math] with respect to a metric which penalizes the closeness to their geodesic midpoint.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2014-06-18 |