Jiuyi Zhu
Louisiana State University, LA·Mathematics & Statistics
A vibrating guitar string or drum skin moves in a set of basic patterns called modes. In mathematics these modes are eigenfunctions of the Laplace operator, a building block of the partial differential equations (PDE) that describe vibration.
- Fields
Mathematics & Statistics
- Topics
partial differential equations
Laplacian eigenfunctions
nodal sets
- Grants
- 1 known grant, through Jul 2028
What they work on
- Estimate the size of zero-level (nodal) sets of Laplacian eigenfunctions for second-order and higher-order elliptic equations, and determine how that size depends on high frequency.
- Study level sets of eigenfunctions in periodic elliptic homogenization, where equation coefficients vary in a fine repeating pattern.
- Prove quantitative Cauchy uniqueness and propagation of smallness for elliptic equations with singular weights, measured on measurable sets of positive measure.
- Apply quantitative unique continuation results to null controllability for heat equations and to energy density estimates in spectral theory.
More about the research
Eigenfunctions are higher-dimensional versions of the familiar sine and cosine functions on a circle. As the frequency grows, their shapes and the places where they concentrate change. A central question is how those changes depend on frequency, and how the size of a vibration in one region limits its size elsewhere. These questions drive the quantitative study of solutions of PDE.
Why it matters
This is basic mathematics. It sharpens understanding of how solutions of vibration and diffusion equations behave at high frequency and how their size propagates through space. The resulting estimates could inform control theory, including how heat can be steered to a desired state, and spectral theory, which studies the characteristic frequencies of a system.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Shiferaw Berhanu
University of Maryland, MD·Mathematics & Statistics
Determine the geometric conditions on two embedded CR manifolds M and M' under which a CR mapping that vanishes to infinite order at a point must vanish identically near that point.
William Minicozzi
Massachusetts Institute of Technology, MA·Mathematics & Statistics
Establish gradient estimates and monotonicity formulas for harmonic functions and heat equation solutions, clarifying how they depend on dimension and how elliptic and parabolic behavior relate.
Simon Bortz
University of Alabama, AL·Mathematics & Statistics
Characterize parabolic uniform rectifiability of time-varying boundaries through properties of caloric measure, the distribution of heat flowing in from the boundary, and through estimates for bounded solutions of the heat equation.
The school
Louisiana State University: 15 labs on PI Pages, $431M a year on research, 323 research doctorates a year.
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