Welcome to JIUYI ZHU's Webpage
Address:
Jiuyi Zhu
224 Lockett Hall
Baton Rouge, LA 70803
Email: zhu at math dot lsu dot edu
Appointment:
o Associate Professor (August 2022 - Present)
Research Interests:
o
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.
o CV
Publications and preprints
- Upper bounds of nodal sets for solutions of bi-Laplace
equations II, arXiv:2603.04677
- Quantitative unique continuation for Neumann problem in
planar C^{1,\alpha} domains, arXiv:2512.07297
(with Y. Cai and J. Zhuge).
- 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, arXiv:2505.03996
(with E. Malinnikova),
- 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
o Nodal sets of Dirichlet eigenfunctions in quasiconvex Lipschitz domains, arXiv:2303.02046 (with J. Zhuge).
o Spectral inequality for Schrodinger equations with power growth potentials, arXiv:2301.12338 (with J. Zhuge), To appear in Indiana University Mathematics Journal
o
Upper
bounds of critical sets of elliptic equations on the plane, Vietnam Journal
of Mathematics, 51(2023),799-810.
o 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
o Doubling inequalities and upper bounds of critical sets of Dirichlet eigenfunctions, Journal of Functional analysis. 281(2021), no. 8, 109155. Link
o Upper bounds of nodal sets for eigenfunctions of eigenvalue problems, Mathematische Annalen, (2022) 382:1957-1984.(with F.H. Lin). Link
o Propagation of smallness in elliptic periodic homogenization, SIAM Journal on Mathematical Analysis, 53(2021), no.1,111-132, Link.(with C. Kenig)
o
Boundary doubling
inequality and nodal sets nodal sets of Robin and Neumann eigenfunctions, Potential analysis, 59 (2023),375–407. Link,
o
Doubling inequality and
nodal sets for solutions of bi-Laplace equations, Archive for Rational
Mechanics and Analysis
,232(2019), no.3,1543-1595.Link
o
Fractional equations with indefinite
nonlinearities, Discrete & Continuous Dynamical Systems, Series A, 39(2019),
no.3, 1257-1268.(with W. Chen and C. Li). Link
o
Quantitative uniqueness of solutions to parabolic
equations, Journal of Functional analysis. 275(2018), no. 9,
2373-2403. Link
o
Quantitative unique continuation of solutions to
higher order elliptic equations with singular coefficients, Calc.Var. & PDE 57:58(2018), Link
o
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
o 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
o Geometry and interior nodal sets of Steklov eigenfunctions, Calculus of Variations and Partial Differential Equations, 59(2020), no. 5, 150. Link
o Interior nodal sets of Steklov eigenfunctions on surfaces,Analysis & PDE, 9(2016), no. 4, 859-880. Link
o Lower bounds for interior nodal sets of Steklov eigenfunctions, Proceedings of the AMS, 144(2016), 4715-4722 (With Chris Sogge and X. Wang). Link
o A lower bound for nodal sets of Steklov eigenfunctions, Mathematical Research Letters, 22(2015), 1243-1253 (with X. Wang). Link
o Doubling property and vanishing order of Steklov eigenfunctions, Communications in Partial Differential Equations, 40(2015), no. 8, 1498-1520. Link
o Indefinite fractional elliptic problem and Liouville theorems, J. Differential Equations, 260(2016), 4758-4785 (with W. Chen). Link
o Quantitative uniqueness of elliptic equations, American Journal of Mathematics, 138(2016), no. 3, 733-762. Link
o Maximum principles and symmetry results of viscosity solutions for fully nonlinear equations, J. Differential Equations 258(2015), 2054-2079 (with G. Lu). PDF
o The improved Moser-Trudinger inequality with L^p norm in n dimensions, Advanced Nonlinear Studies 14(2014), no. 2, 273-293. PDF
o Liouville-type theorems for fully nonlinear elliptic equations in half spaces, Advanced Nonlinear Studies 13(2013), 979-1001 (with G. Lu). PDF
o 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
o 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
o An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Analysis 75(2012), no. 6, 3036-3048 (with G. Lu). PDF
o 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
o 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
o 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
o 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
o 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
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