6533b825fe1ef96bd1281f0f
RESEARCH PRODUCT
Regular 1-harmonic flow
Salvador MollMichał ŁAsicaMichał ŁAsicaLorenzo Giacomellisubject
Applied Mathematics010102 general mathematicsMathematical analysisBoundary (topology)Total variation flow; harmonic flow; well-posednessRiemannian manifoldLipschitz continuitySubmanifold01 natural sciencesManifoldDomain (mathematical analysis)35K51 35A01 35A02 35B40 35D35 35K92 35R01 53C21 68U10010101 applied mathematicsMathematics - Analysis of PDEsFlow (mathematics)FOS: MathematicsMathematics::Differential GeometrySectional curvature0101 mathematicsAnalysisAnalysis of PDEs (math.AP)Mathematicsdescription
We consider the 1-harmonic flow of maps from a bounded domain into a submanifold of a Euclidean space, i.e. the gradient flow of the total variation functional restricted to maps taking values in the manifold. We restrict ourselves to Lipschitz initial data. We prove uniqueness and, in the case of a convex domain, local existence of solutions to the flow equations. If the target manifold has non-positive sectional curvature or in the case that the datum is small, solutions are shown to exist globally and to become constant in finite time. We also consider the case where the domain is a compact Riemannian manifold without boundary, solving the homotopy problem for 1-harmonic maps under some assumptions.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2017-11-20 | Calculus of Variations and Partial Differential Equations |