Concentration sets of the Landau�Lifshitz system and quasi-mean curvature flows

Xian-Gao Liu
December 2006
Calculus of Variations & Partial Differential Equations;Dec2006, Vol. 27 Issue 4, p493
Academic Journal
The aim of this work is to analyze the concentration set of the stationary weak solutions to the Landau-Lifshitz system of the ferromagnetic spin chain from Rm into the unit sphere S� of R�. Suppose that uk ? u weakly in W1,2(Rm � R+, S�) and that St is the concentration set for fixed t. In the present paper we first prove that St is a Hm-2-rectifiable set for almost all t ? R+. And then we verify that St moves by the quasi-mean curvature under some assumptions, which is a new codimension 2 curvature flow. Finally we analyze the behavior of the solution at the singular point and get the blow up formulas. The main barrier to Landau-Lifshtiz system is that there is no energy monotonicity inequality. After the seminal work to on the study of the concentration set of minimizing energy harmonic maps by Leon Simon, there are several papers dealing with the stationary harmonic maps and its heat flows, and so on. This investigation is inspired by the study on the heat flow of harmonic maps and it largely depends on our result of the partial regularity.


Related Articles

  • On the Kolmogorov Problem for the Upper Bounds of Four Consecutive Derivatives of a Multiply Monotone Function. Babenko, Vladislav; Babenko, Yuliya // Constructive Approximation;Jun2007, Vol. 26 Issue 1, p83 

    In this paper we shall give necessary and sufficient conditions for the system of positive numbers $M_{k_0}, M_{k_1}, M_{k_2}, M_{k_3}, 0=k_0

  • A singular limit arising in combustion theory: Fine properties of the free boundary. Weiss, G.S. // Calculus of Variations & Partial Differential Equations;Jul2003, Vol. 17 Issue 3, p311 

    The equation ?tue-?ue = -?e(ue) in (0,&infinity;)�Rn, where ?e converges to the Dirac measure concentrated at 0 with mass 1/2, has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in C0,1(Rn) and have a uniformly bounded...

  • A Concept of Monotonicity and Its Characterization for Closed Queueing Networks. Suri, Rajan // Operations Research;May/Jun85, Vol. 33 Issue 3, p606 

    We define a qualitative property, called monotonicity, for a closed queueing network. This property is defined both for a station in the network and for the network as a whole. Several results in this paper relate monotonic behavior to the known parameters of a network and allow us to state a...

  • Subdifferential representations of risk measures. Pflug, Georg Ch. // Mathematical Programming;Sep2006, Vol. 108 Issue 2/3, p339 

    Measures of risk appear in two categories: Risk capital measures serve to determine the necessary amount of risk capital in order to avoid ruin if the outcomes of an economic activity are uncertain and their negative values may be interpreted as acceptability measures (safety measures). Pure...

  • On the method of spiral arms monotony to determine the orientation angles of galaxies. Poltorak, S.; Fridman, A. // Astronomy Reports;Jun2007, Vol. 51 Issue 6, p460 

    A method is proposed for the determination of the position and inclination angles of the plane of a spiral galaxy based on the assumption that every spiral arm is a monotonic function of the radius and/or the azimuthal angle. This method may yield more accurate results than the more commonly...

  • LIMITED MEMORY BFGS METHOD FOR NONLINEAR MONOTONE EQUATIONS. Weijun Zhou; Donghui Li // Journal of Computational Mathematics;Jan2007, Vol. 25 Issue 1, p89 

    In this paper, we propose an algorithm for solving nonlinear monotone equations by combining the limited memory BFGS method (L-BFGS) with a projection method. We show that the method is globally convergent if the equation involves a Lipschitz continuous monotone function. We also present some...

  • Bounds for how much influence an observation can have. Olkin, Ingram; Raveh, Adi // Statistical Methods & Applications;2009, Vol. 18 Issue 1, p1 

    That outliers or influential observations can affect the results in a regression is well-known. But it is not clear how much influence a specific observation can have on other statistics. In time series, especially in predictive situations, the effect of additional observations is of singular...

  • Drop out monotonic rules for sequencing situations. Fernández, Cristina; Borm, Peter; Hendrickx, Ruud; Tijs, Stef // Mathematical Methods of Operations Research;2005, Vol. 61 Issue 3, p501 

    This note introduces a new monotonicity property for sequencing situations. A sequencing rule is called drop out monotonic if no player will be worse off whenever one of the players decides to drop out of the queue before processing starts. This intuitively appealing property turns out to be...

  • One-way monotonicity as a form of strategy-proofness. Sanver, M. Remzi; Zwicker, William S. // International Journal of Game Theory;2009, Vol. 38 Issue 4, p553 

    Suppose that a vote consists of a linear ranking of alternatives, and that in a certain profile some single pivotal voter v is able to change the outcome of an election from s alone to t alone, by changing her vote from P v to $${P^\prime_{v}}$$ . A voting rule $${\mathcal{F}}$$ is two-way...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics