| NSF Org: |
OIA OIA-Office of Integrative Activities |
| Recipient: |
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| Initial Amendment Date: | August 20, 2018 |
| Latest Amendment Date: | August 20, 2018 |
| Award Number: | 1832961 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Subrata Acharya
acharyas@nsf.gov (703)292-2451 OIA OIA-Office of Integrative Activities O/D Office Of The Director |
| Start Date: | October 1, 2018 |
| End Date: | September 30, 2022 (Estimated) |
| Total Intended Award Amount: | $166,612.00 |
| Total Awarded Amount to Date: | $166,612.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
5801 S Ellis Ave Chicago IL US 60637-5418 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): |
ANALYSIS PROGRAM, EPSCoR Research Infrastructure |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.083 |
ABSTRACT
Nontechnical Description
Mathematics provides essential tools for the description and investigation of numerous real-life problems. Rate of change, which is the derivative in mathematics, is fundamental in these problems. Partial differential equations (PDEs) are differential equations that relate unknown multivariable functions and their partial derivatives. They describe a wide variety of seemingly distinct physical phenomena such as sound, heat, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics. Because of this, PDEs play a prominent role in many disciplines including engineering, physics, economics, and biology. One of the central problems in the study of PDEs is about uniqueness and its quantitative properties. The fellowship builds a collaboration between Louisiana State University (LSU) and University of Chicago (UChi) by enabling the PI to make extended research visits to UChi. This project will lead to advancements in understanding uniqueness of PDEs, stimulate the PI's research capacity, strengthen the research program in PDEs and analysis at LSU, and benefit its undergraduate and graduate education.
Technical Description
The goal of this research is to investigate the quantitative and qualitative properties of uniqueness of solutions to PDEs. Providing quantitative and qualitative information for the solutions is essential in the study of PDEs, which lies in the core of mathematical analysis. Often the most effective way to obtain such information is to first explore the quantitative and qualitative properties of solutions of the equations and then to develop algorithms in accordance; this is a beneficial alternative to the challenge of instead solving PDEs computationally with sufficient accuracy. Quantitative uniqueness, a recent fast-developing area, describes the quantitative behavior of the strong unique continuation property. The proposed research is to study the quantitative uniqueness for elliptic PDEs with singular weights. The outcome will provide a deeper understanding of the strong unique continuation property for this category of PDEs. It is important and interesting to study how exactly the information on a small open set propagates to any other open sets in the domain, which quantifies the weak unique continuation property. Such exploration will lead to many applications in inverse problems and control theory. The PI will explore how the coefficient functions determine the uniqueness and how the shape of the domain influences the uniqueness of solutions for parabolic PDEs. The project will contribute the fundamental understanding of elliptic and parabolic partial differential equations.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
The project is concerned with quantitative uniqueness of solutions of partial differential equations. Specifically, the PI studied the quantitative unique continuation property and nodal sets for solutions of partial differential equations. Providing quantitative and qualitative information for the solutions is essential to the study of partial differential equations, which lies in the core of mathematical analysis.
During the grant period, the PI visited the host at University of Chicago in the spring of 2019 and 2020. The PI, with the collaborators, have done those important contributions to the field. (1) the PI with the collaborator developed new quantitative Carleman estimates to study the vanishing order for solutions of elliptic equations with singular potentials; (2) The PI made significant contributions to the upper bounds of boundary nodal sets for Neumann and Robin eigenfunctions. The PI also obtained sharp upper bound for boundary critical sets for Dirichlet eigenfunctions; (3) The PI obtained the sharp upper bound for interior nodal sets of Steklov eigenfunctions. The PI has shown the doubling inequalities and finite upper bounds of nodal sets for solutions of bi-Laplace equations; (4) Homogenization theory aims to understand the properties of materials with complicated microstructures. The PI with the collaborators obtained almost three-ball inequality in periodic elliptic homogenization for the first time. Furthermore, we developed new techniques to obtain the explicit doubling inequalities and upper bounds of nodal sets in elliptic periodic homogenization.
The PI have been actively engaged in outreach activities for K-12 students by making mathematical demonstrations with hands-on activities in various outreach events and supervising middle school students for science fair projects. The PI has mentioned undergraduate students and one graduate student for their research projects. The PI has disseminated the results obtained in this award via the publications in high quality journals and delivered many lectures in conferences and seminars. Those activities have benefited a broader community of STEM workforce.
Last Modified: 11/01/2022
Modified by: Jiuyi Zhu
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