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A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

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Science China Mathematics 2021年第7期64卷 1391-1410页

作者： lu Chen guozhen lu Maochun Zhu School of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081China Department of Mathematics University of ConnecticutStorrsCT 06269USA School of Mathematical Sciences Institute of Applied System AnalysisJiangsu UniversityZhenjiang 212013China

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)lutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential *** a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential *** existence result seems to be the first one without this standard assumption.

关键词： Trudinger-Moser inequalities degenerate potential ground state solutions Schrodinger equations Nehari manifold

On second order minimal Lagrangian submanifolds in complex space forms

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Science China Mathematics 2005年第11期48卷 1505-1516页

作者： DONG Yuxin lu guozhen Institute of Mathematics Fudan University Shanghai 200433 China Department of Mathematics Wayne State University Detroit MI48202 USA

In this paper, we determine all second order minimal Lagrangian submanifolds in complex space forms whose cubic forms have the largest non-trivial continuous symmetries. We describe these minimal Lagrangian submanifol... 详细信息

关键词： exterior differential systems,minimal Lagrangian submanifold

Hardy–Littlewood–Sobolev Inequalities with the Fractional Poisson Kernel and Their Applications in PDEs

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Acta Mathematica Sinica,English Series 2019年第6期35卷 853-875页

作者： lu CHEN guozhen lu Chunxia TAO School of Mathematics and Statistics Beijing Institute of Technology Department of Mathematics University of Connecticut School of Mathematical Sciences Beijing Normal University

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel on the upper half space ■ where f ∈ L^p(?R_+~n), g ∈ Lq(R_+~n) and p, q'∈(1, +∞), 2 ≤α lutions to the integral system of the Euler–Lagrange equations associated with the extremals of the fractional Poisson kernel. Finally, by using the method of moving plane in integral forms, we prove that extremals of the Hardy–Littlewood–Sobolev inequality with the fractional Poisson kernel must be radially symmetric and decreasing about some point ξ_0 ∈ ?R_+~n. Our results proved in this paper play a crucial role in establishing the Stein–Weiss inequalities with the Poisson kernel in our subsequent paper.

关键词： Existence of extremal functions Hardy–Littlewood–Sobolev inequality Moving plane method Poisson kernel

Polynomials,Higher Order Sobolev Extension Theorems and Interpolation Inequalities on Weighted Folland-Stein Spaces on Stratified Groups

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Acta Mathematica Sinica,English Series 2000年第3期16卷 405-444页

作者： guozhen lu Department of Mathematics,Wayne State University,Detroit,MI 48202,USA Department of Mathematics Wayne State University Detroit USA

This paper consists of three main *** of them is to develop local and global Sobolev interpolation inequalities of any higher order for the nonisotropic Sobolev spaces on stratified nilpotent Lie *** the extensive research after Jerison’s work[3]on Poincaré-type inequalities for Hrmander’s vector fields over the years,our results given here even in the nonweighted case appear to be *** interpolation inequalities have crucial applications to subelliptic or parabolic PDE’s involving vector *** main tools to prove such inequalities are approximating the Sobolev func- tions by polynomials associated with the left invariant vector fields on *** very useful properties for polynomials associated with the functions are given here and they appear to have independent interests in their own *** the existence of such polynomials is the second main part of this *** results of these two parts have been announced in the author’s paper in Mathematical Research Letters[38]. The third main part of this paper contains extension theorems on anisotropic Sobolev spaces on stratified groups and their applications to proving Sobolev interpolation inequalities on(εδ)domains. Some results of weighted Sobolev spaces are also given *** construct a linear extension operator which is bounded on different Sobolev spaces *** particular,we are able to construct a bounded linear extension operator such that the derivatives of the extended function can be controlled by the same order of derivatives of the given Sobolev *** are stated and proved for weighted anisotropic Sobolev spaces on stratified groups.

关键词： Poincaré inequalities Extension theorems Interpolation inequalities Anisotropic Sobolev spaces A_p weights (ε,δ)domains Vector fields Polynomials on stratified groups

Special Issue in Honor of Professor Carlos E. Kenig

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Acta Mathematica Sinica,English Series 2019年第6期35卷 719-720页

作者： guozhen lu Zhongwei Shen University of Connecticut University of Kentucky

It is our great honor and pleasure to be the guest editors for this special issue of the Acta Mathematica Sinica-English Series, a flagship journal of the Chinese Mathematical Society, dedicated to Professor Carlos E.... 详细信息

关键词： Kenig Special Issue in Honor of Professor Carlos E

EMBEDDING THEOREMS ON CAMPANATO-MORREY SPACES FOR VECTOR FIELDS OF HoRMANDER TYPE

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Analysis in Theory and Applications 1998年第1期14卷 69-80页

作者： guozhen lu (Wright State University, USA) Department of Mathematics and Statistics Wright State University Dayton USA

This paper deals with embedding theorems on Campanato-Marrey spaces formed by degenerate vector fields, which include Honnander and Grushin type of vector fields. These embedding theorems are somewhat different from the known Poincare estimates. The main ingredients of the proofs rely on the fractional maximal functions. These results evidently have applications to the regularity of subelliptic PDE.

关键词： EMBEDDING THEOREMS ON CAMPANATO-MORREY SPACES FOR VECTOR FIELDS OF HoRMANDER TYPE Math

Study on the optical system for the record of the pulsed acoustic field and the scattering of the cylindrical-like objects buried in underwater sand

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Chinese Journal of Acoustics 2002年第3期21卷 193-200页

作者： ZHU guozhen lu Ke’an YANG Xuhui FU Deyong(Department of Physics, Tsinghua University Beijing 100084)

The response of the two working mode of the optical systems, the schlieren mode and shadowgraph mode, for taking of optical photograph of the pulsed acoustic field in liquid are studied. It shows that, the response of the optical intensity on the screen to the acoustic pressure is linear for shadowgraph mode and nonlinear for the schlieren mode. Because the function of shadowgraph mode has no limit on working frequency, it is suitable for the studies of the acoustic field of laboratory model of the seabed or the buried objects at low ultrasonic range. The ultrasonic pulse scattered by the cylindrical-like objects buried in underwater sand was studied experimentally by the shadowgraph mode at low ultrasonic frequency. There are five kinds of the scattered waves from the half-buried object and three kinds of the waves from the full-buried objects were recorded. The two kinds of creeping waves (the longitudinal wave and the shear wave of the object) appear in both of the two cases.

关键词： mode Study on the optical system for the record of the pulsed acoustic field and the scattering of the cylindrical-like objects buried in underwater sand

Loss of TET1 in oligodendrocyte evokes psychiatric disorders in the mice

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Loss of TET1 in oligodendrocyte evokes psychiatric disorders...

Techniques and physical properties of 10MHz short pulse focused ultrasonic transducer

中国神经科学学会第十二届全国学术会议

作者： Ming Zhang guozhen lu Wenjie Guo Shengxi Wu Xianghui Zhao Department of Neurosciences and Collaborative Innovation Center for Brain Science the Fourth Military Medical University

The significance of myelin defects in psychiatric disorders has been revealed in recent studies, suggesting that oligodendrocyte（OL） abnormalities underlie some aspects of these diseases. 5-hydroxymethylcytosine（5 hm C）, an intermediate of DNA demethylation, and its dioxygenases, TET family members, have been identified as new epigenetic players in the regulation of nervous system development, as well as cognition, memory and other important neural functions. In previous study, we have shown the important function of Tet1 in controlling the normal development of OLs and myelination. Here, we examined a possible role for OL expressed TET1 in several brain functions, such as anxiety, depression, cognition and learning under a battery of behavior tests. Selective knockout of TET1 in OL linage did not change the locomotor activity in open field test, but the TET1 mutant spent more time and travelled longer distance in the central area of the field, indicating their anti-anxiety like behaviors. However, these mice were not defective in the elevated plus maze test, which was not in accordance with their behavior in the open field. Meanwhile, the mutant mice showed similar performance to their littermates in novel object recognition and social interaction tests, which suggest their normal cognitive function. In tail suspension test, a validated method for examining depression like behavior, TET1 mutant showed increased immobility duration than the control mice. Finally, following five days of T maze training, we found that TET1 mutant mice performed worse than controls, exhibiting decrease accuracy in alternating their choice of goal arm. These findings reveal a novel role for TET1, an enzyme important for DNA hydroxymethylation, in the related neural functions associated with hypomyelination, and provide new insight into the pathophysiology of psychiatric disorders, which can aid in future drug discovery efforts.

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声学技术 2004年第S1期23卷 15-17,22页

作者： ZHU guozhen, YANG Yong, lu Kean (Department of Physics, Tsinghua University, Beijing 100084, China) Department of Physics Tsinghua University Beijing 100084 China

A focused ultrasonic transducer used for biomedical purposes with a fundamental frequency of 10MHz and a pulse width of one and a half periods is described in this paper. Its physical properties are given including (1) focused acoustic field recorded by an optical means, (2) electric waveform for triggering the transducer and the corresponding waveform of the wave received by another transducer, and (3) result of tests on a sample object.

关键词： focused transducer short pulse ultrasonic medicine nondestructive testing