Heat flow ofp-harmonic maps with values into spheres

Chen, Yunmei; Hong, Min-Chun; Hungerbühler, Norbert

doi:10.1007/bf02571698

Cited by 74 publications

(77 citation statements)

References 12 publications

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“…Hungerbühler [14] established existence of global weak solutions of the p-harmonic flow between Riemannian manifolds M and N for arbitrary initial data having finite p-energy in the case when the target N is a homogeneous space with a left invariant metric when 2 < p < n. Chen-Hong-Hungerbühler [8] proved existence of global weak solutions when p ≥ 2.…”

Section: Introductionmentioning

confidence: 99%

REGULARITY CRITERIA FOR THE p-HARMONIC AND OSTWALD-DE WAELE FLOWS

Fan¹,

Nakamura²,

Zhou³

2015

Bulletin of the Korean Mathematical Society

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Section: Introductionmentioning

confidence: 99%

REGULARITY CRITERIA FOR THE p-HARMONIC AND OSTWALD-DE WAELE FLOWS

Fan¹,

Nakamura²,

Zhou³

2015

Bulletin of the Korean Mathematical Society

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“…[11,12,13,16,18,21,22,30,32,31,34,38,39,40,41,46,48] for 1 < p < ∞; [26,27] for p = 1). In the case when the target manifold is a sphere, the existence of a global weak solution for the harmonic flow was first proved by Chen in [11] using a penalization technique.…”

Section: Introductionmentioning

confidence: 99%

“…The result and the penalization technique were extended to the p-harmonic flow for p > 2 by Chen, Hong and Hungerbühler in [12]. The p-harmonic flow for 1 < p < 2 was solved by Misawa in [41] using a time discretization technique (the method of Rothe) proposed in [31], and by Liu in [39] using a penalization technique similar to that of [12]. The p-harmonic flow (1 < p < ∞) from a unit ball in R m into S 1 ⊂ R 2 was studied by Courilleau and Demengel in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, our theory handles the p-harmonic map heat flow (1.8)-(1.11) for all 1 ≤ p < ∞ in a unified fashion, rather than treating the system separately for different values of p and using different methods (cf. [11,12,13,16,18,21,22,30,31,34,41,46,48]). Secondly, based on the above theoretical work, we also develop and analyze a practical fully discrete finite element method for approximating the solutions of the p-harmonic map heat flow (1.8)-(1.11).…”

Section: Introductionmentioning

confidence: 99%

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

Barrett¹,

Feng²,

Prohl³

2008

SIAM J. Math. Anal.

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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.

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“…The equation (1.1) with (1.4) also concerns the negative gradient flow for p-harmonic maps between smooth, compact Riemannian manifolds (cf. [4,17,14] and, for p = 2, see [8,10,19]), and the smallness condition (1.7) implies a geometric relation between the curvature of the target manifold and the image of a solution (see [9]). …”

mentioning

confidence: 99%