MA 773, Introduction to Geometric Measure Theory
Courses taught previously:
Spring 2007
- MA214, Calculus IV
- MA641G, Differential Geometry
Fall 2007
- MA213, Calculus III
- MA575, Principle of Mathematical Analysis
Spring 2008
- MA114, Calculus II
- MA 676, Analysis I
Fall 2008
- MA 213, Calculus III
- MA 677, Analysis II
PDE Seminar, Spring 2009 (Time:11-12:00pm Tuesday; Room: POT 745)
Previous analysis/pde seminar
Previous Publications (Partially supported by NSF grants):
Recent Publications (The research
reported in the following papers is supported by NSF grants
DMS 0400718 and DMS 0601162)
- Stable stationary harmonic maps into spheres.
Acta Math. Sinica (Engl. Ser.) 22 (2006), no. 2, 319-330
(with FangHua Lin). pdf file.
- On Landau-Lifshitz equation in dimensions at most four.
Indiana Univ. Math. J. Vol. 55, no.6 (2006) 1615-1644.
dvi file.
- Generalized cone comparison principle for viscosity solutions of
the Aronsson equation and absolute minimizers. Comm. P.D.E. V. 31, No.
7 (2006) 1027-1046 (with R. Gariepy, Y. Yu). dvi file
.
- The Aronsson equation for absolute minimizers of $L^\infty$
functionals associated with vector fields satisfying H\"ormander's
conditions. Trans. AMS. 359 (2007), 91-113.
dvi file.
- Heat flow of biharmonic maps in dimension four and its
application. Pure and Applied Math. Quarterly, Vol. 3, No. 2 (2007)
595-613 (Special issue: In honor of L. Simon, Part 1 of 2). pdf file.
- Finite time singularity of the Landau-Lifshitz-Gilbert equation.
Int. Math. Reas. Notices, Vol. 2007 (with Shijin Ding). pdf file
- Asymptotic behavior of infinity harmonic functions near an
isolated singularity. Int. Math.
Reas. Notices, Vol. 2008, article ID rnm163, 23 pages (with O. Savin,
Y. Yu). pdf file .
- Heat flow of harmonic maps whose gradients belong to
$L^n_xL^\infty_t$. Arch. Rational Mech. Anal. 188 (2008), 309-349.
pdf file
- Harmonic maps from manifolds of $L^\infty$-Riemannian metrics.
Calc. Var. & PDE (2008), Vol 32, 387-405 (with W. Ishizuka). pdf
file .
- $C^1$-regularity of the Aronsson equation in $\R^2$,
Annales I'Institut H. Poincar\'e- Analyse non lin\'eaire, 25 (2008) 659-678
(with Y. Yu). dvi file .
- Uniqueness and nonuniqueness of viscosity solutions to Aronsson's
equation. Arch. Ration Mech. Anal. (with
R. Jensen, Y. Yu), Vol. 190, no. 2 (2008) 347-370. pdf file .
- Derivation of the Aronsson equation for C-1 Hamiltonians.
Trans. AMS 361 (2009) 103-124(with M. Crandall, Y. Yu). pdf file .
- Aronsson's equation on Carnot-Carath\'eodory spaces.
To appear, Illinois Journal of Mathematics (with Y. Yu). pdf
file .
- Boundary regularity for polyharmonic maps in the critical dimension.
To appear, Advances in Calculus of Variations (with T. Lamm).
pdf file .
- An open question on regularity of Aronsson's equation.
Communications in Mathematical Analysis, Proceeding of the 2006 Midwest Geometry
Conference, pp. 80-83 (2008), edited by E. B. Lin, W. Wei.
pdf file
- A Remark on Nonlinear Dirac Equations.
To appear, Proceedings of American Mathematical Society
pdf file .
- Regularity of Dirac-harmonic maps.
To appear, Int. Math. Reas. Notices, (with D. L. Xu).
pdf file .
- Landau-Lifshitz-Maxwell equation in dimension three.
To appear, Pacific J. Math. (with S.J. Ding, X.G. Liu).
pdf file .
Preprints
- Notes on the regularity of harmonic map systems.
Preprint (2009) (with T. Huang)
pdf file
- Liquid crystall flows in two dimensions.
Preprint (2009) (with F. H. Lin, J. Y. Lin)
pdf file
- On the Keller-Rubinstein-Sternberg problem, I.
In preparation (with F. H. Lin, X. B. Pan).
- Global solutions to the compressible liquid crystal flows in one dimension.
Preprint (2009) (with S. J. Ding, J. Y. Lin, Y. H. Wen).
- Partial regularity of suitable weak solutions to the liquid crystal flow in three dimensions.
In preparation (with F. H. Lin).
Monography
- The Analysis of Harmonic Maps and Their Heat Flows, 267 pages
(with F. H. Lin), by World Scientific Publisher Inc, ISBN-109812779523.
- An Introduction of Infinity Harmonic Functions.
Preprint, 2008. pdf file .
