
RESEARCH INSTERESTS

I.
Harmonic Analysis on Manifolds:
Detailed study of the relationship between the growth estimates ($L^p$, bilinear, multilinear,
and gradient estimates) of the eigenfunctions and the global geometric properties on
compact manifolds.
Apply the eigenfunction estimates to study the location, distribution and size of nodal
sets of eigenfunctions, and to study H\"ormander multiplier problems, BochnerRiesz means
for eigenfunction expansion on compact manifolds.
Apply the eigenfunction estimates for spectral projectors on manifolds (with or without
boundary) to study wellposedness problems for partial differential equations on compact
manifolds, including linear or nonlinear wave equations, Schr\"odinger equations, 2D (dissipative)
quasigeostrophic equations, and 2D Euler equations.

II.
Nonlinear differential equations:
Study LiYau type sharp differential Harnack inequalities, the heat kernel estimates,
and the monotonicity of entropy for linear heat equations and Schr\"odinger operators on
Riemannian manifolds with negative Ricci curvature.
Study Liouville's Theorems for Schr\"odinger operators on Riemannian manifolds with
nonnegative Ricci curvature.
Study gradient estimates for degenerate parabolic equations and Liouville's Theorems,
local AronsonBenilan estimates and entropy formulae for Porous Media Equations and
Fast Diusion Equations.
Study the global uniqueness problems and the boundary stabilization, controllability
and observability problems for (linear and nonlinear) parabolic and hyperbolic PDE's on
manifolds via Carleman estimates.
Study
the Periodic solutions, subharmonics and homoclinic orbits of
Hamiltonian systems.

THESIS

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)


Xiangjin Xu, Subharmonic solutions
of a class of nonautonomous Hamiltonian systems. Acta Sci.
Nat. Univer. Nankai. Vol. 32, No.2, (1999), pp. 4650.(In
Chinese)
Yiming Long, Xiangjin Xu,
Periodic solutions for a class of nonautonomous Hamiltonian
systems. Nonlinear
Anal. Ser. A: Theory Methods, 41
(2000), no. 34, 455463. (PDF)
Xiangjin Xu, Homoclinic orbits for
first order Hamiltonian systems possessing superquadratic
potentials. Nonlinear
Anal. Ser. A: Theory Methods, 51
(2002), no. 2, 197214. (PDF)
Xiangjin Xu, Periodic solutions for
nonautonomous Hamiltonian systems possessing superquadratic
potentials. Nonlinear
Anal. Ser. A: Theory Methods, 51
(2002), no. 6, 941955. (PDF)
Xiangjin Xu, Subharmonics for first
order convex nonautonomous Hamiltonian systems. J.
Dynam. Differential Equations 15
(2003), no. 1, 107123. (PDF)
Xiangjin Xu, Multiple solutions of
superquadratic second order dynamical systems. Dynamical systems
and differential equations (Wilmington, NC, 2002). Discrete
Contin. Dyn. Syst. 2003,
suppl., 926934. (PDF)
Xiangjin Xu, Subharmonics 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, 643654. (PDF)
Xiangjin Xu, Homoclinic orbits for
first order Hamiltonian systems with convex potentials. Advanced
Nonlinear Studies 6
(2006), 399410. (PDF)
Xiangjin Xu, New Proof of
H\"ormander Multiplier Theorem on Compact manifolds without
boundary. Proc.
Amer. Math. Soc. 135
(2007), 15851595.(PDF)
Roberto
Triggiani, Xiangjin Xu, Pointwise Carleman Estimates, Global Uniqueness,
Observability, and Stabilization for Schrodinger Equations on
Riemannian Manifolds at the $H^1$Level. AMS
Contemporary Mathematics,
Volume 426, 2007, 339404. (PDF)
Xiangjin Xu, Gradient estimates for
eigenfunctions of compact manifolds with boundary and the
H\"ormander multiplier theorem. Forum
Mathematicum 21:3
(May 2009), pp. 455476. (PDF)
Xiangjin Xu, Eigenfunction
estimates for Neumann Laplacian on compact manifolds with boundary
and multiplier problems. Proc. Amer. Math. Soc. 139 (2011),
35833599.(PDF)
Junfang Li, Xiangjin Xu, Differential Harnack inequalities on Riemannian
manifolds I : linear heat equation.Advance in Mathematics, Volume
226, Issue 5, (March, 2011) Pages 44564491
doi:10.1016/j.aim.2010.12.009
(arXiv:0901.3849
)
Liangui Wang, Xiangjin Xu,
Hybrid state feedback, robust $H_{\infty}$ control for a class
switched systems with nonlinear uncertainty.
Z. Qian et al.(Eds.):Recent Advances in CSIE 2011,
Lecture Notes in Electrical Engineering, Volume 129, 2012, pp 197202
Xiangjin Xu, Gradient estimates for
$u_t=\Delta F(u)$ on manifolds and some Liouvilletype theorems.
Journal of Differential Equation (2011)
doi:10.1016/j.jde.2011.08.004
arXiv:0805.3676
Xiangjin Xu, Upper and lower bounds
for normal derivatives of spectral clusters of Dirichlet
Laplacian. Journal of Mathematical Analysis and Applications,
Volume 387, Issue 1, (March, 2012), Pages 374383
doi:10.1016/j.jmaa.2011.09.003
, ArXiv:1004.2517


Xiangjin Xu, Spectral Expansions of
Piecewise Smooth Functions on compact Riemannian manifolds with
boundary. (preprint)()
Xiangjin Xu,
Multiple periodic solutions of superquadratic Hamiltonian systems with bounded forcing
terms. (preprint)()
Junfanf Li, Xiangjin Xu,New Perelman type LYH differential Harnack inequalities and entropy
formulas for linear heat equations.(Preprint) ()
Xiangjin Xu, New heat kernel estimates on Riemannian manifolds with negative curvature. (Preprint)
()
Xiangjin Xu, Differential Harnack inequalities and a Liouville Type Theorem for the Schr\"ordinger
Operator. (Preprint) ()
Xiangjin Xu, Periodic and subharmonic solutions of Hamiltonian systems possessing "superquadratic"
potentials. (in preparation)()
Junfang Li, Xiangjin Xu, Differential Harnack inequalities on Riemannian manifolds II: Schr\"odinger
operator. (in preparation)()
Xiangjin Xu, Gradient estimates for spectral clusters and Carleson measures on compact manifolds
with boundary. (in preparation)()
Huichao Chen, Xiangjin Xu, Power analysis of a lefttruncated normal mixture distribution with
applications in red blood cell velocities. Presentation (by H. Chen) at Joint Statistical Meetings (JSM),
Montreal, August, 2013. (Manuscript in preparation)()

My research is
partially supported by the NSF Grant NSFDMS
0602151(June 1 2006November 30, 2008) and NSFDMS0852507
(June 1, 2008May 31, 2010), and partially supported by Harpur
College Grant in Support of Research, Scholarship and Creative
Work in Year 20102011, 20122013.
