people:xxu:xxu-personal
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people:xxu:xxu-personal [2025/06/30 21:23] xxu |
people:xxu:xxu-personal [2025/11/06 12:41] (current) xxu |
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| <UL> | <UL> | ||
| <LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | <LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | ||
| - | 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, Bochner-Riesz means for eigenfunction expansion on compact manifolds. | + | Detailed study of the spectral theory of elliptic operators (Laplace operator and Schrödinger operator) on compact or complete manifolds, in particular, on the growth estimates (Lp, bilinear, and gradient estimates) of eigenfunctions, multiplier problems, Carleson measures and Logvinenko-Sereda sets on compact or complete manifolds with or without boundary. |
| - | </FONT></FONT></P><LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | + | |
| - | Apply the eigenfunction 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, Schr\"odinger equations, 2D (dissipative) quasi-geostrophic equations, and 2D Euler equations. | + | |
| </FONT></FONT></P> | </FONT></FONT></P> | ||
| </UL> | </UL> | ||
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| <TD WIDTH=991 VALIGN=TOP> | <TD WIDTH=991 VALIGN=TOP> | ||
| - | <P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 16pt"><B>II. Nonlinear differential equations: </B></FONT> | + | <P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 16pt"><B>II. Geometric PDEs: </B></FONT> |
| </P> | </P> | ||
| <UL> | <UL> | ||
| <LI><P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 14pt"> | <LI><P ALIGN=LEFT><FONT SIZE=4 STYLE="font-size: 14pt"> | ||
| - | Study Li-Yau 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. | + | Li-Yau and Hamilton type gradient estimates, sharp estimates for the heat kernel and the Green's function for heat equations and Schrödinger operators on Riemannian manifolds (Finsler manifolds, metric measure spaces). Gradient estimates, Liouville's Theorems and entropy formulae for linear and nonlinear (possible degenerate) parabolic equations. Control theoretic problems for (linear and nonlinear) parabolic and hyperbolic PDE systems on manifolds via Carleman estimates. Periodic solutions, subharmonics and homoclinic orbits of Hamiltonian systems. |
| - | </FONT></P> | + | |
| - | <LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | + | |
| - | Study gradient estimates for degenerate parabolic equations and Liouville's Theorems, local Aronson-Benilan estimates and entropy formulae for Porous Media Equations and Fast Diffusion Equations. | + | |
| - | </FONT></P> | + | |
| - | <LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | + | |
| - | 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. | + | |
| - | </FONT> | + | |
| - | </P> | + | |
| - | <LI><P><FONT SIZE=4 STYLE="font-size: 14pt"> | + | |
| - | Study the Periodic solutions, subharmonics and homoclinic orbits of Hamiltonian systems. | + | |
| </FONT></FONT></P> | </FONT></FONT></P> | ||
| </UL> | </UL> | ||
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| <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | ||
| - | <B>Xiangjin Xu</B>, Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary. Springer Proceedings in Mathematics & Statistics, vol 471. Springer, Cham. 2024. <A HREF="https://doi.org/10.1007/978-3-031-69706-7_1">https://doi.org/10.1007/978-3-031-69706-7_1</A>(<A HREF="Xu-Carleson.pdf"></A>)</FONT></P> | + | <B>Xiangjin Xu</B>, Characterization of Carleson Measures via Spectral Estimates on Compact Manifolds with Boundary. Springer Proceedings in Mathematics & Statistics, vol 471. Page 1-23, Springer,2024. <A HREF="https://doi.org/10.1007/978-3-031-69706-7_1">https://doi.org/10.1007/978-3-031-69706-7_1</A>(<A HREF="Xu-Carleson.pdf"></A>)</FONT></P> |
| </OL> | </OL> | ||
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| <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | ||
| - | <B>Xiangjin Xu</B>, Sharp Hamilton's Gradient and Laplacian Estimates on noncompact manifolds. (Submitted) | + | <B>Xiangjin Xu</B>, Sharp Hamilton's Gradient and Laplacian Estimates on noncompact manifolds. (Submitted April 2025) |
| (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | ||
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| <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | ||
| - | <B>Xiangjin Xu</B>, Heat kernel and Green's function on manifolds with nonnegative Ricci curvature. (Submitted) | + | <B>Xiangjin Xu</B>, Heat kernel and Green's function on manifolds with nonnegative Ricci curvature. (Submitted May 2025) |
| (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | ||
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| <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | <LI><P><FONT SIZE=4 STYLE="font-size: 16pt"> | ||
| - | Xing Wang,<B>Xiangjin Xu</B>, Cheng Zhang, $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds. preprint. | + | Xing Wang,<B>Xiangjin Xu</B>, Cheng Zhang, $L^p$-Logvinenko-Sereda sets and $L^p$-Carleson measures on compact manifolds. arXiv:2506.22759 [math.AP]. (Submitted July 2025) |
| (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | (<A HREF="Xu-HeatKernel-II.pdf"></A>)</FONT></P> | ||
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| <TD WIDTH=991 VALIGN=TOP> | <TD WIDTH=991 VALIGN=TOP> | ||
| <UL> | <UL> | ||
| - | <P><FONT SIZE=4 STYLE="font-size: 16pt">My research is | + | <P><FONT SIZE=4 STYLE="font-size: 16pt">My research is partially supported by:</P> |
| - | partially supported by <B>the NSF Grant <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0602151">NSF-DMS | + | |
| - | 0602151</A>(June 1 2006-November 30, 2008) and <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0852507">NSF-DMS-0852507</A> | + | <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0602151">NSF-DMS 0602151</A>(2006 - 2008) and <A HREF="http://www.nsf.gov/awardsearch/showAward.do?AwardNumber=0852507">NSF-DMS-0852507</A> |
| - | (June 1, 2008-May 31, 2010)</B>, and partially supported by </FONT><FONT FACE="Times New Roman, serif"><FONT SIZE=4 STYLE="font-size: 18pt"><B>Harpur | + | (2008 - 2010)</B>, </P> |
| - | College Grant in Support of Research, Scholarship and Creative | + | |
| - | Work in Year 2010-2011, 2012-2013, 2018-2019.</B></FONT></FONT></P> | + | <P><B>Harpur College Grants in Support of Research, Scholarship and Creative Work:</B> Year 2010-2011, Year 2012-2013, Year 2017-2018, Year 2019-2020.</P> |
| + | |||
| + | <P><B> NYS/UUP Individual Development Awards:</B> Year 2013-2014.</P> | ||
| + | |||
| + | <P><B> AMS-NSF Travel grants:</B> ICM 2010 in Hyderabad, India, Augest 2010. PIMS conference, UBC, Canada, July 2013. The Second PRIMA Congress, Shanghai, China, June 2013. MCA 2021 (Online), July, 2021. MCA 2025, Miami, July, 2025.</P> | ||
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| + | </FONT></FONT></P> | ||
| </UL> | </UL> | ||
| </TD> | </TD> | ||
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| </TABLE> | </TABLE> | ||
| <HR> | <HR> | ||
| - | <P>Last updated: 05/01/2015 | + | <P>Last updated: 07/01/2025 |
| </P> | </P> | ||
| </BODY> | </BODY> | ||
| </HTML> | </HTML> | ||
people/xxu/xxu-personal.1751333026.txt · Last modified: 2025/06/30 21:23 by xxu