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

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, 7-10 junio, 1999.

[4]. J. F. Debongnie and H. Nguyen-Xuan and H. C. Nguyen. Dual analysis for finite element solutions of plate bending, Proceedings of the Eighth International Conference on Computational Structures Technology, Civil-Comp Press, Stirlingshire, Scotland, 2006.

[5]. H. Nguyen-Xuan 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 sub-supersolutions 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 p-Laplacian, 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 Bidaut-Ve ́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.