Regular 1-harmonic flow

Abstract

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.

A local estimate for vectorial total variation minimization in one dimension

“…In view of (12), up to a subsequence u ε converges weakly in H 1 (I) and uniformly in C(I) to the unique minimizer u ∈ H 1 (I) of E (Γ-convergence of E ε to E is straightforward). From (13) we see that…”

“…We also expect that a similar technique can be used to prove analogous versions of the results when the range of u is constrained to a Riemannian submanifold in R n . In fact, the existence theory for constrained total variation flows (also known as 1-harmonic flows) is at present limited to non-generic targets [14,15,11] (not including the orientation space SO(3)) and Lipschitz initial data [16,13]. We believe that an a priori estimate of form (7) will provide a convenient tool for generalizing the existence theory to the case of initial data of bounded variation into generic target manifolds.…”

A new numerical scheme for discrete constrained total variation flows and its convergence

“…In [20], the existence of local-in-time regular solution was proved when Ω is the k-torus T k , the manifold M is an ( −1)-sphere S −1 and the initial datum u 0 is sufficiently smooth and of small total variation. Recently, this work has been improved significantly in [15]. In particular, the assumption has been weakened to convex domain Ω and Lipschitz continuous initial data u 0 .…”

“…In particular, the assumption has been weakened to convex domain Ω and Lipschitz continuous initial data u 0 . Moreover, in [15], the existence of global-in-time regular solution and its uniqueness have been proved when the target manifold M has non-positive curvature, and the initial datum u 0 is small.…”

Why capillary flows in slender triangular grooves are so stable against disturbances

“…To gain further insight into the growth or decay of disturbances at finite times, it is useful to consider the discretized operator exp(Lτ ) in diagonalized form such that exp(Lτ ) = S exp(Λτ )S −1 (53) where exp(Λτ ) denotes a diagonal matrix whose entries are the eigenvalues of L arranged in decreasing order and S represents the matrix whose columns are the corresponding eigenfunctions. It can then be shown [45] that…”