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Some Weighted Norm Inequalities on Manifolds

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Chinese Annals of Mathematics,Series B 2018年第6期39卷 1001-1016页

作者： Shiliang ZHAO school of Mathematical Sciences Fudan University

Let M be a complete non-compact Riemannian manifold satisfying the volume doubling property and the Gaussian upper bounds. Denote by A the Laplace-Beltrami operator and by V the Riemannian gradient. In this paper, the author proves the weighted reverse inequality ｜｜Δ^1/2f｜｜L^p（w）×C｜｜｜Δ↓｜｜｜L^p（w）, for some range ofp determined by M and w. Moreover, a weak type estimate is proved when p = 1. Some weighted vector-valued inequalities are also established.

关键词： Weighted norm inequality Poincare inequality Riesz transform.

Exact Controllability with Internal Controls for First-Order Quasilinear Hyperbolic Systems with Zero Eigenvalues

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Chinese Annals of Mathematics,Series B 2016年第4期37卷 503-514页

作者： Kaili ZHUANG school of Mathematical Sciences Fudan University

For first-order quasilinear hyperbolic systems with zero eigenvaiues, the author establishes the local exact controllability in a shorter time-period by means of internal controls acting on suitable domains. In particular, under certain special but reasonable hypotheses, the local exact controllability can be realized only by internal controls, and the control time can be arbitrarily small.

关键词： First-order quasilinear hyperbolic system Zero eigenvalue Exactcontrollability Internal control

Rankin-Cohen Deformations and Representation Theory

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Chinese Annals of Mathematics,Series B 2014年第5期35卷 817-840页

作者： Yijun YAO school of Mathematical Sciences Fudan University

The author uses the unitary representation theory of SL2(R) to understand the Rankin-Cohen brackets for modular forms. Then this interpretation is used to study the corresponding deformation problems that Paula Cohen,... 详细信息

关键词： Modular forms Rankin-Cohen brackets Representation theory Rankin-Cohen deformation

Cyclic Covers over Strongly Liftable Schemes

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Chinese Annals of Mathematics,Series B 2015年第6期36卷 937-946页

作者： Qihong XIE school of Mathematical Sciences Fudan University

A smooth scheme X over a field k of positive characteristic is said to be strongly liftable over W_2(k),if X and all prime divisors on X can be lifted simultaneously over W_2(k).In this paper,the author gives a criterion for those cyclic covers over strongly liftable schemes that are still strongly *** a corollary,cyclic covers over projective spaces of dimension at least three are strongly liftable over W_2(k).

关键词： Cyclic cover Strongly liftable scheme Positive characteristic

Co-Poisson structures on polynomial Hopf algebras

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Science China Mathematics 2018年第5期61卷 813-830页

作者： Qi Lou Quanshui Wu school of Mathematical Sciences Fudan University

The Hopf dual H~? of any Poisson Hopf algebra H is proved to be a co-Poisson Hopf algebra provided H is noetherian. Without noetherian assumption, unlike it is claimed in literature, the statement does not hold. It is proved that there is no nontrivial Poisson Hopf structure on the universal enveloping algebra of a non-abelian Lie algebra. So the polynomial Hopf algebra, viewed as the universal enveloping algebra of a finite-dimensional abelian Lie algebra, is considered. The Poisson Hopf structures on polynomial Hopf algebras are exactly linear Poisson structures. The co-Poisson structures on polynomial Hopf algebras are *** correspondences between co-Poisson and Poisson structures are also established.

关键词： Poisson algebra co-Poisson coalgebra Poisson Hopf algebra co-Poisson Hopf algebra

Global existence and stability of a stationary Mach reflection

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Science China Mathematics 2015年第1期58卷 11-34页

作者： CHEN Shu Xing school of Mathematical Sciences Fudan University

We prove the global existence and stability of a wave structure containing a stationary Mach con- figuration, which occurs when an incident shock front hits a wall with a large incident angle. Our result shows that tile data of the upstream flow and the pressure at downstream part jointly determine the whole flow, as well a the wave structure. Particularly, we show that the height of the Mach stem depends not only on the data of upstream flow, but also on the pressure at downstream flow. The flow with the assigned wave structure is governed by a free boundary value problem for the Euler system. In the problem the location of the triple point, the shock fronts and the contact discontinuity are all unknown, they are finally determined together with the solution.

关键词： Euler system Mach reflection existence stability free boundary problem elliptic-hyperboliccomposcd system

Fundamental solution method for inverse source problem of plate equation

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Applied Mathematics and Mechanics(English Edition) 2012年第12期33卷 1513-1532页

作者：顾智杰谭永基 school of Mathematical Sciences Fudan University

The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method （FSM） and the Tikhonov regularization technique axe used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

关键词： Kirchhoff-Love plate Euler-Bernoulli beam elastic inverse source problem,fundamental solution method (FSM) Tikhonov regularization method meshless numericalmethod

Maximum principle for optimal control of neutral stochastic functional differential systems

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Science China Mathematics 2015年第6期58卷 1265-1284页

作者： WEI WenNing school of Mathematical Sciences Fudan University

This paper is concerned with optimal control of neutral stochastic functional differential equations(NSFDEs). The Pontryagin maximum principle is proved for optimal control, where the adjoint equation is a linear neutral backward stochastic functional equation of Volterra type(VNBSFE). The existence and uniqueness of the solution are proved for the general nonlinear VNBSFEs. Under the convexity assumption of the Hamiltonian function, a sufficient condition for the optimality is addressed as well.

关键词： neutral stochastic functional differential equation neutral backward stochastic functional equationof Volterra type stochastic optimal control Pontryagin maximum principle

Local time and Tanaka formula of G-martingales

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Applied Mathematics(A Journal of Chinese Universities) 2019年第4期34卷 468-479页

作者： LIU Guo-min school of Mathematical Sciences Fudan University

The objective of this paper is to study the local time and Tanaka formula of symmetric *** introduce the local time of G-martingales and show that it belongs to the G-expectation space LG^2(ΩT).By a localization argument,we obtain the bicontinuous modification of local ***,we give the Tanaka formula for convex functions of G-martingales.

关键词： G-martingale Local time Tanaka formula