Welcome to JIUYI  ZHU's Webpage   

                        

                                                           

         Address:

             Jiuyi Zhu                                              

             Louisiana State University

             Department of Mathematics

             224 Lockett Hall

             Baton Rouge, LA 70803

　

　

            Email: zhu at math dot lsu dot edu

                                                            

     

　

　

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Appointment:

• Associate Professor (August 2022 - Present)

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Research Interests:

•  Partial differential equations and their applications, harmonic analysis, spectral theory and geometric analysis. Some studied research topics: Geometric properties of eigenfunctions, Quantitative uniqueness of solutions of PDEs, Elliptic Homogenization, Qualitative properties of solutions of nonlinear PDEs, Geometric inequalities, etc.
   

•  CV

　

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 Publications and preprints

• Quantitative propagation of smallness for elliptic equations with singular potentials, (with E. Malinnikova), preprint
• Spectral inequalities for Schr\"odinger equations and quantitative propagation of smallness in the plane, (with E. Malinnikova), preprint
• Boundary quantitative unique continuation for solutions of elliptic equations, arXiv:2409.15198 (with J. Dalberg)
• Observability inequalities for heat equations with potentials, arXiv:2409.09476, (with J. Zhuge)
• Spectral inequalities for Schrödinger equations with various potentials, arXiv:2403.08975

• Nodal sets of Dirichlet eigenfunctions in quasiconvex Lipschitz domains, arXiv:2303.02046 (with J. Zhuge).

• Spectral inequality for Schrodinger equations with power growth potentials, arXiv:2301.12338 (with J. Zhuge), To appear in Indiana University Mathematics Journal

• Upper bounds of critical sets of elliptic equations on the plane, Vietnam Journal of Mathematics, 51(2023),799-810.

Link

• Doubling inequalities and nodal sets in periodic elliptic homogenization, Communications in Partial Differential Equations, 47 (2022), no. 3, 549-584(with C. Kenig and J. Zhuge). Link

• Doubling inequalities and upper bounds of critical sets of Dirichlet eigenfunctions, Journal of Functional analysis. 281(2021), no. 8, 109155. Link

• Upper bounds of nodal sets for eigenfunctions of eigenvalue problems, Mathematische Annalen, (2022) 382:1957-1984.(with F.H. Lin). Link

• Propagation of smallness in elliptic periodic homogenization, SIAM Journal on Mathematical Analysis, 53(2021), no.1,111-132, Link.(with C. Kenig)

• Boundary doubling inequality and nodal sets nodal sets of Robin and Neumann eigenfunctions, Potential analysis,  59 (2023),375–407. Link,

• Doubling inequality and nodal sets for solutions of bi-Laplace equations, Archive for Rational Mechanics and Analysis

,232(2019), no.3,1543-1595.Link

• Fractional equations with indefinite nonlinearities, Discrete & Continuous Dynamical Systems, Series A, 39(2019), no.3, 1257-1268.(with W. Chen and C. Li). Link

• Quantitative uniqueness of solutions to parabolic equations, Journal of Functional analysis. 275(2018), no. 9, 2373-2403. Link

• Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients, Calc.Var. & PDE 57:58(2018), Link

• Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions,Calc. Var.& PDE 57:92 (2018)(with B. Davey),Link

• Quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms, Communications in Partial Differential Equations, 4(2019), no.11,1217-1251(with B. Davey), Link

•  Geometry and interior nodal sets of Steklov eigenfunctions, Calculus of Variations and Partial Differential Equations, 59(2020), no. 5, 150. Link

•  Interior nodal sets of Steklov eigenfunctions on surfaces,Analysis & PDE,  9(2016), no. 4, 859-880. Link

•  Lower bounds for interior nodal sets of Steklov eigenfunctions, Proceedings of the AMS, 144(2016), 4715-4722 (With Chris Sogge and X. Wang). Link

•  A lower bound for nodal sets of Steklov eigenfunctions, Mathematical Research Letters, 22(2015), 1243-1253 (with X. Wang). Link

•  Doubling property and vanishing order of Steklov eigenfunctions, Communications in Partial Differential Equations, 40(2015), no. 8, 1498-1520. Link

•  Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations,  260(2016), 4758-4785 (with W. Chen). Link

•  Quantitative uniqueness of elliptic equations, American Journal of Mathematics, 138(2016), no. 3, 733-762. Link

•  Maximum principles and symmetry results of viscosity solutions for fully nonlinear equations,  J. Differential Equations 258(2015), 2054-2079 (with G. Lu). PDF

•  The improved Moser-Trudinger inequality with L^p norm in n dimensions, Advanced Nonlinear Studies 14(2014), no. 2, 273-293. PDF  

•  Liouville-type theorems for fully nonlinear elliptic equations in half spaces, Advanced Nonlinear Studies 13(2013), 979-1001 (with G.  Lu). PDF

•  Liouville-type theorems and decay estimates for solutions to higher order elliptic equations, Ann.Inst.H.Poincare Anal.Non Lineaire,29(2012), no.5, 653-665(with  G. Lu and P. Wang). PDF

•  Hardy-Littlewood-Sobolev and Stein-Weiss inequalities on the Heisenberg group, Nonlinear Analysis 75(2012), no. 11 4296-4314 (with X. Han and G. Lu).    PDF

•  An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis 75(2012), no. 6, 3036-3048 (with G. Lu). PDF

•  The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific Journal of Mathematics 253(2011), no. 2, 455-473 (with  G. Lu). PDF

•  Characterization of balls in terms of Bessel-potential integral equations, J. Differential Equations 252(2012), no. 2, 1589-1602 (with X. Han and G. Lu). PDF

•  A priori estimates, existence and non-existence of positive solutions of generalized mean curvature equations, Nonlinear Analysis 74(2011), no.18, 7126-7136(with Q. Dai and Y. Gu). PDF

• Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality, Calc. Var. & PDE. 42(2011), no.3-4, 563-577 (with G. Lu). PDF

•  Radial symmetry and regularity of solution for poly-Harmonic Dirichlet problems, J. Math. Anal. Appl. 377(2011), no.2, 744-753 (with W. Chen). PDF

　

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Some Outreach activities.

 

1, Demonstration of Chladni Pattern,  Photo1, Photo 2, Photo 3
 

2, Classroom presentations to K-12 girls, Photo.

 

3, Undergraduate research projects,

• Vibration of Drumhead: Shapes in the Music, PDF

• Nodal Patterns in Vibrations, PDF

4. Middle school student's science and engineer fair project

• Isoperimetric Inequality, PDF