Xiangjin Xu

Assistant Professor Department of Mathematical Sciences at Binghamtom University Ph.D. 2004, Johns Hopkins University. Thesis advisor: Christopher D. Sogge At Binghamton since 2007.

Office: LN 2226
Phone: (607)777-2514
FAX: (607)777-2450
E-mail: xxu@math.binghamton.edu
Address: Department of Mathematical Science, Binghamton University, Vestal Parkway East, Binghamton, NY 13902-6000

MY Brief Curriculum Vitae

Research Interests:

1. Harmonic Analysis on Manifolds:

Eigenfunction estimates and multiplier problems on Riemannian manifolds, Gibbs' phenomenon and Pinsky's phenomenon for Fourier inversion and eigenfunction expansion.

2. Nonlinear differential equations:

Well-posedness problems for nonlinear hyperbolic differential equations on manifolds; Boundary stabilization, controllability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on manifolds; Periodic solutions, subharmonics and homoclinic orbits of Hamiltonian systems.

Publications and Preprints

Master Thesis: Periodic solutions of Hamiltonian systems and differential systems. Nankai Institute of Mathematics, Tianjin, China, June 1999.

PhD Thesis: Eigenfunction Estimates on Compact Manifolds with Boundary and H\"ormander Multiplier Theorem. Johns Hopkins University, Baltimore, Maryland, May 2004.(PDF)

1. Subharmonic solutions of a class of non-autonomous Hamiltonian systems. Acta Sci. Nat. Univer. Nankai. Vol. 32, No.2, (1999), pp. 46-50.(In Chinese)

2. (with Yiming Long) Periodic solutions for a class of nonautonomous Hamiltonian systems. Nonlinear Anal. 41 (2000), no. 3-4, Ser. A: Theory Methods, 455-463. (PDF)

3. Homoclinic orbits for first order Hamiltonian systems possessing super-quadratic potentials. Nonlinear Anal. 51 (2002), no. 2, Ser. A: Theory Methods, 197-214. (PDF)

4. Periodic solutions for non-autonomous Hamiltonian systems possessing super-quadratic potentials. Nonlinear Anal. 51 (2002), no. 6, Ser. A: Theory Methods, 941-955. (PDF)

5. Subharmonics for first order convex nonautonomous Hamiltonian systems. J. Dynam. Differential Equations 15 (2003), no. 1, 107-123. (PDF)

6. Multiple solutions of super-quadratic second order dynamical systems. Dynamical systems and differential equations (Wilmington, NC, 2002). Discrete Contin. Dyn. Syst. 2003, suppl., 926-934. (PDF)

7. Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Nonlinear differential equations, mechanics and bifurcation (Durham, NC, 2002). Discrete Contin. Dyn. Syst. Ser. B 3 (2003), no. 4, 643-654. (PDF)

8. Homoclinic orbits for first order Hamiltonian systems with convex potentials. Advanced Nonlinear Studies 6 (2006), 399-410. (PDF)

9. New Proof of H\"ormander Multiplier Theorem on Compact manifolds without boundary. Proc. Amer. Math. Soc. 135 (2007), 1585-1595.(PDF)

10. (with Roberto Triggiani) Pointwise Carleman Estimates, Global Uniqueness, Observability, and Stabilization for Schr¨odinger Equations on Riemannian Manifolds at the $H^1$-Level. AMS Contemporary Mathematics, Volume 426, 2007, 339-404. (In Control Methods in PDE Dynamical Systems, AMS-IMS-SIAM Joint Summer Research Conference, July 3-7, 2005 Snowbird, Utah, Edited by F. Ancona, I. Lasiecka, W. Littman and R. Triggiani.) (PDF)

11. Gradient estimates for eigenfunctions of Riemannian manifolds with boundary. (preprint)

12. Gradient estimates for eigenfunctions of compact manifolds with boundary and the H\"ormander multiplier theorem. (Submitted)

13. Spectral Expansions of Piecewise Smooth Functions on compact Riemannian manifolds with boundary. (preprint)

14. Gradient estimates for spectral cluster with C^{1,1} metrics and multiplier problems. (in preparation)

15. Eigenfunction estimates for Neumann Laplacian on compact manifolds with boundary and multiplier problems. (in preparation)

My research is partially supported by the NSF Grant (NSF-DMS 0602151): Eigenfunction Estimates on Manifolds with Boundary and Applications to Partial Differential Equations, June 1st, 2006 - May 31st, 2009.