Approximation of elliptic and parabolic equations with Dirichlet boundary conditions

Youchan Kim; Seungjin Ryu; Pilsoo Shin; Youchan Kim; Seungjin Ryu; Pilsoo Shin

doi:10.3934/mine.2023079

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-43. doi: 10.3934/mine.2023079

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Approximation of elliptic and parabolic equations with Dirichlet boundary conditions

1.
Department of Mathematics, University of Seoul, 163 Seoulsiripdaero, Dongdaemun-gu, Seoul 02504, Republic of Korea
2.
Department of Mathematics, Kyonggi University, 154-42 Gwanggyosan-ro, Yeongtong-gu, Suwon 16227, Republic of Korea

Received: 22 August 2022 Revised: 08 January 2023 Accepted: 08 March 2023 Published: 20 March 2023

We obtain an approximation result of the weak solutions to elliptic and parabolic equations with Dirichlet boundary conditions. We show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains.
- nonlinear elliptic equations,
- nonlinear parabolic equations,
- approximations
Citation: Youchan Kim, Seungjin Ryu, Pilsoo Shin. Approximation of elliptic and parabolic equations with Dirichlet boundary conditions[J]. Mathematics in Engineering, 2023, 5(4): 1-43. doi: 10.3934/mine.2023079

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Abstract

We obtain an approximation result of the weak solutions to elliptic and parabolic equations with Dirichlet boundary conditions. We show that the weak solution can be obtained with a limit of approximations by regularizing the nonlinearities and approximating the domains.

References

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