New regularization and error estimate on terminal value problem for elliptic equations

Nguyen Anh Triet; Danh Hua Quoc Nam; Nguyen Huu Can; Le Dinh Long

doi:10.3934/dcdss.2023110

Article Contents

2024, Volume 17, Issue 3: 1028-1048. Doi: 10.3934/dcdss.2023110

This issue Previous Article On a non-local Kirchhoff type equation with random terminal observation Next Article Dynamics of locally monotone stochastic evolution equations

New regularization and error estimate on terminal value problem for elliptic equations

1.
Department of Mathematics, University of Architecture Ho Chi Minh City (UAH) 196 Pasteur Str., Dist. 3, Ho Chi Minh City, Vietnam
2.
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
3.
Vietnam National University, Ho Chi Minh City, Vietnam
4.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
5.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
6.
Falculty of Maths, FPT University HCM, Saigon Hi-tech Park, Ho Chi Minh City, Vietnam

^*Corresponding author: Nguyen Huu Can (nguyenhuucan@tdtu.edu.vn)
^*Corresponding author: Nguyen Huu Can (nguyenhuucan@tdtu.edu.vn)

Received: February 2023

Revised: April 2023

Early access: September 4, 2023

Published online: September 4, 2023

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Abstract

In this paper, we study the terminal value problem for elliptic equation with various forms of source functions. This equation has many applications in the fields of physics, mechanics, electromagnetism. Until now, there are not many studies that focus on the regularization in $ L^{p} $ spaces. Our results are the first work on the inverse problem for elliptic equations in $ L^p $. The principal technique is to use Fourier regularized solution combined with Sobolev embeddings. The error between the exact and regularized solutions are obtained in $ L^{p} $ under some suitable assumptions on the Cauchy data.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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