Onp-Harmonic Map Heat Flows for $1\leqp<\infty$ and Their Finite Element Approximations

Abstract

Abstract. Motivated by emerging applications from imaging processing, the heat flow of a generalized p-harmonic map into spheres is studied for the whole spectrum, 1 ≤ p < ∞, in a unified framework. The existence of global weak solutions is established for the flow using the energy method together with a regularization and a penalization technique. In particular, a BV -solution concept is introduced and the existence of such a solution is proved for the 1-harmonic map heat flow. The main idea used to develop such a theory is to exploit the properties of measures of the forms A · ∇v and A ∧ ∇v; which pair a divergence-L 1 , or a divergence-measure, tensor field A, and a BV -vector field v. Based on these analytical results, a practical fully discrete finite element method is then proposed for approximating weak solutions of the p-harmonic map heat flow, and the convergence of the proposed numerical method is also established.

Very singular diffusion equations: second and fourth order problems

“…The theory is further developed by [36]. Global existence of weak solutions is recently established by Barrett et al [7]. However, its uniqueness is not known even locally (except in the one dimensional case, see Giga and Kobayashi [56]).…”

The 1-Harmonic Flow with Values into $\mathbb S^{1}$

“…x∈ Ω (here ν denotes the outward unit normal to ∂Ω), and an existence result is presented for that notion [6,Theorem 6.12]. The proof of this result is discussed in section 7: we argue that its correctness is questionable (see Remark 7.6 and Example 7.7) and that the argument is confined to data and solutions with empty jump set (see Proposition 7.4 and Remark 7.5).…”

“…(similar to [6,19]), or the (1 + ε)-harmonic flow. In any such approximation, the wedge product with u ε cancels the right-hand side (r.h.s.…”

Divergence-L q and divergence-measure tensor fields and gradient flows for linear growth functionals of maps into the unit sphere

“…Firstly, we shall introduce a BV -solution concept and present a weak theory for the parabolic system (10)- (13). To the best of our knowledge, besides the recent work of [5] which developed a global weak theory in the case ϕ(s) = s and the work of [18] which studied the local solvability of the case ϕ(s) = s, no result is known in the literature for the gradient flow into spheres for general linear growth functionals. Secondly, based on our theoretical result, we also develop and analyze some practical fully discrete finite element methods for computing the solutions of the gradient flow (10)- (13).…”

“…As in [5,[8][9][10]21,24,28], the key ideas for passing to the limit are to exploit the symmetries of the unit sphere S n−1 and to use the following compactness result, which can be proved following the proof of Theorem 2.1 of [9] (also see Theorem 3 of Chap. 4 of [15]) using the fact that the operator defined by the first three terms on the left-hand side of (20) is uniformly parabolic.…”