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On the morning of 19 January, at VNUHCM-University of Science, doctoral researcher Lưu Xuân Thắng, specialising in Applied Mathematics, successfully defended his doctoral thesis entitled “The Cauchy Problem for Elliptic Equations with Noisy Coefficients” 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 focuses on two central issues: the existence of weak solutions to the problem within appropriate functional spaces, and the identification of suitable regularisation methods together with an analysis of the convergence of the regularised solutions towards the exact solution.

Drawing on tools from real analysis, partial differential equations, and inverse problem theory, the study applies the Liouville transformation to reduce variable-coefficient problems to constant-coefficient forms. Appropriate functional spaces are then systematically employed to establish the existence of weak solutions. Regularised solutions are obtained using filtering methods and Fourier truncation techniques, followed by rigorous estimates to analyse their convergence. Finally, illustrative examples and numerical results are presented to support and validate the theoretical findings.

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, and investigating a similar problem for wave equations and higher-order wave equations.

Minh Tâm _ Translated by ℙ𝕄ℕ