Harmonic and quasi-harmonic spheres, part III. Rectifiablity of the parabolic defect measure and generalized varifold flows

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 19, Issue 2, 2002, Pages 209-259

  Lin, FangHua ^a   Wang, ChangYou ^b  

^a NEW YORK UNIVERSITY   (United States)

^b UNIVERSITY OF KENTUCKY   (United States)

Author keywords

Brakke's flow; Concentration measures; Energy quantization; Generalized varifold flows; Harmonic or approximated harmonic map flows; Rectifiablity; Stratification

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; HARMONIC ANALYSIS; PROBLEM SOLVING; SET THEORY; THEOREM PROVING;

BRAKKE'S FLOW; ENERGY QUANTIZATION; QUASI-HARMONIC SPHERES; STRATIFICATION; VARIFOLD FLOWS;

NONLINEAR EQUATIONS;

EID: 0347985182     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0294-1449(01)00090-7     Document Type: Article

References (38)

1. Ambrosio L., Soner H.M. A measure-theoretic approach to higher codimension mean curvature flows (Dedicated to Ennio De Giorgi). Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4). 25:(1-2):1997;27-49. (1998).
2. Ambrosio L., Soner H.M. Level set approach to mean curvature flow in arbitrary codimension. J. Differential Geom. 43:(4):1996;693-737.
3. Almgren F.J. Jr. Q valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two. Bull. Amer. Math. Soc. (N.S.). 8:(2):1983;327-328.
4. Almgren F.J. Jr. The Theory of Varifolds. Mimeographed Notes, Princeton. 1965.
5. Allard W.K. An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled. Geometric Measure Theory and the Calculus of Variations, Arcata, CA, 1984. Proc. Sympos. Pure Math. 44:1986;1-28 American Mathematical Society, Providence, RI.
6. Allard W.K. On the first variation of a varifold. Ann. of Math. (2). 95:1972;417-491.
7. Bethuel F. On the singular set of stationary harmonic maps. Manu. Math. 78:(4):1993;417-443.
8. Brakke K. The Motion of a Surface by its Mean Curvature. Mathematical Notes. 20:1978;Princeton University Press, Princeton, NJ.
9. Cheng X.X. Estimate of the singular set of the evolution problem for harmonic maps. J. Differential Geom. 34:(1):1991;169-174.
10. Chen Y.M., Li J.Y., Lin F.H. Partial regularity for weak heat flows into spheres. Comm. Pure Appl. Math. 48:(4):1995;429-448.
11. Chen Y.M., Lin F.H. Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1:(3-4):1993;327-346.
12. Caffarelli L., Kohn R., Nirenberg L. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math. 35:(6):1982;771-831.
13. Chen Y.M., Struwe M. Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201:(1):1989;83-103.
14. Ding W.Y., Tian G. Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom. 3:(3-4):1995;543-554.
15. Eells J., Sampson J. Harmonic mappings of riemannian manifolds. Amer. J. Math. 86:1964;109-160.
16. Federer H. Geometric Measure Theory. 1969;Springer-Verlag, New York.
17. Feldman M. Partial regularity for harmonic maps of evolution into spheres. Comm. Partial Differential Equations. 19:(5-6):1994;761-790.
18. Federer H., Ziemer W.P. The Lebesgue set of a function whose distribution derivatives are p th power summable. Indiana Univ. Math. J. 22:1972/73;139-158.
19. Helein F. Regularite des applications faiblement harmoniques entre une surface et une variete riemannienne. C. R. Acad. Sci. Paris Ser. I Math. 312:(8):1991;591-596.
20. Ilmanen T. Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature. J. Differential Geom. 38:(2):1993;417-461.
21. Ilmanen T. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc. 108:(520):1994.
22. Jerrard R., Soner H.M. Scaling limits and regularity results for a class of Ginzburg-Landau systems. Ann. Inst. H. Poincaré Anal. Non Lineaire. 16:(4):1999;423-466.
23. Lin F.H. Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. of Math. (2). 149:(3):1999;785-829.
24. Lin F.H. Mapping problems, fundamental groups and defect measures. Acta Math. Sin. (Engl. Ser.). 15:(1):1999;25-52.
25. Lin F.H. Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension- 2 submanifolds. Comm. Pure Appl. Math. 51:(4):1998;385-441.
26. Lin F.H. Varifold type theory for Sobolev mappings. AMS/IP Stud. Adv. Math. 20. 2001;423-430 American Mathematical Society, Providence, RI.
27. Lin F.H., Riviere T., Energy quantization for harmonic maps. Duke Math. J. (to appear).
28. Lin F.H., Riviere T. A quantization property for static Ginzburg-Landau vortices. C.P.A.M. 54:(2):2001;206-228.
29. Li J.Y., Tian G. Blow-up Locus for heat flows of harmonic maps. Acta Math. Sin. (Engl. Ser.). 16:(1):2000;29-62.
30. Lin F.H., Wang C.Y. Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. Partial Differential Equations. 6:(4):1998;369-380.
31. Lin F.H., Wang C.Y. Harmonic and quasi-harmonic spheres. Comm. Anal. Geom. 7:(2):1999;397-429.
32. Lin F.H., Wang C.Y., Harmonic and quasi-harmonic spheres, Part II, Comm. Anal. Geom. (to appear).
33. Simon L. Lectures on geometric measure theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. 1983;Australian National University, Centre for Mathematical Analysis, Canberra.
34. Simon L. Theorems on regularity and singularity of energy minimizing maps. Lectures in Mathematics ETH Zürichür. 1996;Birkhauser Verlag, Basel. (based on lecture notes by Norbert Hungerbhler).
35. Struwe M. On the evolution of harmonic maps in higher dimensions. J. Differential Geom. 28:(3):1988;485-502.
36. Struwe M. On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60:(4):1985;558-581.
37. White B. Stratification of minimal surfaces, mean curvature flows, and harmonic maps. J. Reine Angew. Math. 488:1997;1-35.
38. Ziemer W.P. Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Graduate Texts in Mathematics. 120:1989;Springer-Verlag, New York.