Summer Meeting 2008
Organizers
Bùi Bội Minh Anh (State University of New York at Buffalo, ambuiboi at buffalo.edu)
Lê Hoàng Long (University of Central Arkansas, longl at uca.edu)
Trần Tấn Quốc (University of Wisconsin, Madison, tran at stat.wisc.edu)
Nguyễn Trọng Toán (Indiana University, nguyentt at indiana.edu)
Phan Văn Tuộc (University of British Columbia, phan at math.ubc.edu)
Trương Trung Tuyến (Indiana University, truongt at indiana.edu)
Huỳnh Quang Vũ (ĐHKHTN TPHCM, hqvu at hcmuns.edu.vn)
Time
13 July 2008
Room
F102, ĐHKHTN TPHCM, 227 Nguyễn Văn Cừ, District 5
Speakers
 Lê Dũng (University of Texas at San Antonio)
 Nguyễn Xuân Hùng (Đại học Khoa học Tự nhiên TPHCM)
 Nguyễn Hoàng Lộc (University of Utah)
 Lê Quang Nẫm (New York University)
 Nguyễn Công Phúc (Purdue University)
 Đặng Đức Trọng (Đại học Khoa học Tự nhiên TPHCM)
There is a seminar by Trương Trung Tuyến and short courses by Lê Dũng, Phạm Xuân Dự, Nguyễn Hoài Minh, Nguyễn Công Phúc.
Program
Time 
Speakers 
Abstract 
8:00 ‒ 8:45 
Welcome 

9:00 ‒ 9:50 
Lê Dũng 

10:05 ‒ 10:55 


11:10 ‒ 12:00 
Nguyễn Hoàng Lộc 
On positive solutions of quasilinear elliptic equations 
12:00 ‒ 
Lunch 

13:30 ‒ 14:20 
Nguyễn Công Phúc 

14:35 ‒ 15:15 
Lê Quang Nẫm 
A GammaConvergence Approach to the CahnHilliard Equation 
15:30 ‒ 18:00 
Discussions  
18:30 ‒ 
Dinner 
In attendance there were Lê Tự Quốc Thắng (Georgia Institute of Technology), Nguyễn Hoài Minh (Rutgers Univ.), Nguyễn Lê Lực (Rutgers Univ.).
Abstracts
Nguyễn Xuân Hùng
 A dual analysis for mechanics problems
This report revises briefly the application of the dual analysis concept to elasticity problems. In this method, a same problem is analyzed simultaneously by a displacement and an equilibrium model. The energetic distance between these two models is the sum of both global errors and consequently, an upper bound of each of them. After an exposition of the two models, numerical examples are illustrated, which show the high obtainable accuracy of the method.
Keywords: equilibrium element, conforming element, dual analysis
References:
[1]. Fraeijs de Veubeke B.M. Displacement and equilibrium models in the finite element method, in "Stress analysis", edit.Zienkiewicz O.C. Wiley, London, 1965.
[2]. G.Sander. Applications de la methode des elements finis a la flexion des plaques, Coll.Pub.Fac.Sc.Appli. Univ.of Liège, N015, 1969.
[3]. J.F. Debongnie, P. Beckers. Recent advances in the dual analysis theory, IV Congreso Métodos Numéricos in Ingenieria, Sevilla, 710 junio, 1999.
[4]. J. F. Debongnie and H. NguyenXuan and H. C. Nguyen. Dual analysis for finite element solutions of plate bending, Proceedings of the Eighth International Conference on Computational Structures Technology, CivilComp Press, Stirlingshire, Scotland, 2006.
 [5]. H. NguyenXuan and T. Rabczuk and S. Bordas and J. F. Debongnie. A smoothed finite element method for plate analysis, Computer Methods in Applied Mechanics and Engineering, 197: 1184 – 1203, 2008.
Nguyễn Hoàng Lộc
On positive solutions of quasilinear elliptic equations
In 1981 Peter Hess established a multiplicity result for solutions of boundary value problems for nonlinear perturbations of the Laplace operator. The sufficient conditions given were later shown to be also necessary by Dancer and Schmitt. In this work we show that similar (and slightly more general) results hold when the Laplace operator is replaced by the p− Laplacian.
References
[1] E. Dancer and K. Schmitt, On positive solutions of semilinear elliptic equations, Proceedings of the American Mathematical Society, 101 (1987), pp. 445–452.
[2] P. Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Commun. Partial Differential Equations, 6 (1981), pp. 951–961.
[3] V. K. Le and K. Schmitt, Some general concepts of subsupersolutions for nonlinear elliptic problems, Topological Methods in Nonlinear Analysis, 28 (2006), pp. 87–103.
[4] G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Commun. Partial Differential Equations, 16 (1991), pp. 311–361.
Nguyễn Công Phúc
 Singular quasilinear and Hessian equations and inequalities
We give complete characterizations for the solvability of the following quasilinear and Hessian equations:
−∆p u =σuq+ ω, F k[−u] = σuq+ ω, u≥0
on a domain Ω ⊂ R . Here ∆p is the pLaplacian, F k[u] is the k Hessian, and σ, ω are given nonnegative measurable functions (or measures) on Ω. Our results give a complete answer to a problem posed by BidautVe ́ron in the case σ ≡ 1, and extend earlier results due to Kalton and Verbitsky, Brezis and Cabre ́ for general σ to nonlinear operators.
This talk is based on joint work with Igor E. Verbitsky.