Doctoral Candidate Lưu Xuân Thắng successfully defended his doctoral thesis on “The Cauchy Problem for Elliptic Equations with Noisy Coefficients” at VNUHCM-University of Science. The research focused on regularisation methods to solve ill-posed problems involving noisy coefficients and inaccurate input data, with potential applications in geophysics, fluid mechanics, and cardiology.

On the morning of 19th January, in Room F102 at the VNUHCM-University of Science, Doctoral Candidate Lưu Xuân Thắng, specialising in Applied Mathematics, successfully defended his doctoral thesis on the topic “The Cauchy Problem for Elliptic Equations with Noisy Coefficients.” The thesis was conducted under the supervision of Professor Đặng Đức Trọng.

The thesis investigated the Cauchy problem for elliptic equations in cases where the coefficients are noisy and the input data is inaccurate. These factors rendered the problem ill-posed, requiring the application of regularisation methods to find stable solutions.

 

The thesis focused on investigating the following issues:

• The existence of weak solutions for the problem when considered in the appropriate space.
• The selection of suitable regularisation methods and an investigation into the convergence of the regularised solutions to the exact solution of the problem.

The Doctoral Thesis Committee asked questions of Doctoral Candidate Lưu Xuân Thắng.

Utilising tools from real analysis, partial differential equations, and the inverse problem, the thesis:

• Applied the Liouville transformation to reduce the problem from variable coefficients to a constant coefficient problem.
• Gradually used appropriate spaces to investigate the existence of weak solutions to the problem.
• Employed filtering methods and Fourier truncation techniques to obtain regularised solutions, and then used estimates to investigate the convergence of the regularised solutions to the exact solution.
• Finally, provided examples and numerical results to illustrate the theoretical findings.

Applications and Future Research Directions:
The Cauchy problem for elliptic equations with constant coefficients has numerous practical applications in fields such as geophysics, fluid mechanics, cardiology, bioelectric fields, and non-invasive testing. Furthermore, when the problem is examined with variable and noisy coefficients, the techniques employed can be applied to a class of problems extending from constant coefficients to variable coefficients.

Several issues related to the problem could be investigated in the future, including:

• Examining the Cauchy problem for elliptic equations involving higher-order derivatives.
• Investigating a similar problem for wave equations and higher-order wave equations.

Doctoral Candidate Lưu Xuân Thắng with his supervisor, Professor Đặng Đức Trọng.

Thesis defence photo: 19.01.2025, LVTS Report

Minh Tâm _ Translated by ℙ𝕄ℕ