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Assistant Professor
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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
Study the Lp estimates and gradient estimates on eigenfunctions of Dirichlet (or Neumann) Laplacian, which includes a detailed study of the relationship between the growth estimates of the eigenfunctions and spectrum on the manifolds and global geometric properties.
Apply the Lp estimates and gradients estimates on eigenfunctions to study the location, distribution and size of nodal sets of eigenfunctions.
And apply the Lp estimates and gradient estimates to study Hormander multiplier problems, Bochner-Riesz means for eigenfunction expansion on compact manifolds.
Apply the gradient, bilinear and multilinear estimates for spectral projectors on manifolds (with or without boundary) to study well-posedness problems for partial differential equations on compact manifolds, including linear or nonlinear wave equations, Schrodinger equations, 2D (dissipative) quasi-geostrophic equations, and 2D Euler equations.
Study the boundary stabilization, controllability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on manifolds via Carleman estimates.
Study gradient estimates for degenerate parabolic equations and Liouville's Theorems for Porous Media Equations and Fast Diffusion Equations.
Study Li-Yau type differential Harnack inequalities and the monotonicity of entropy for linear heat equations on Riemannian manifolds with negative Ricci curvature lower bounds.
Study the Periodic solutions, subharmonics and homoclinic orbits of Hamiltonian systems.
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 compact manifolds with boundary and the H\"ormander multiplier theorem. FORUM MATHEMATICUM 21:3 (May 2009), pp. 455-476. (PDF)
12. Spectral Expansions of Piecewise Smooth Functions on compact Riemannian manifolds with boundary. (preprint)
13. Gradient estimates for spectral cluster with $C^{1,1}$ metrics and multiplier problems. (in preparation)
14. Eigenfunction estimates for Neumann Laplacian on compact manifolds with boundary and multiplier problems. (in preparation)
15. Gradient estimates for the degenerate parabolic equation $u_t=\Delta F(u)$ on manifolds and some Liouville theorems of Porous Media Equations. arXiv:0805.3676
16. ( with Junfang Li) Differential Harnack inequalities on Riemannian manifolds I : linear heat equation.arXiv:0901.3849
17. (With Junfang Li) NEW PERELMAN TYPE LYH DIFFERENTIAL HARNACK INEQUALITIES AND ENTROPY FORMULAS FOR LINEAR HEAT EQUATIONS.(in preparation)