NON-SPURIOUS SOLUTIONS OF DISCRETE MIXED BOUNDARY VALUE PROBLEM WITH SINGULAR <i>ϕ</i>-LAPLACIAN

Man Xu; Ruyun Ma; Ting Wang

doi:10.11948/20220455

2023 Volume 13 Issue 4

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Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(4): 2250-2266

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Man Xu, Ruyun Ma, Ting Wang. NON-SPURIOUS SOLUTIONS OF DISCRETE MIXED BOUNDARY VALUE PROBLEM WITH SINGULAR ϕ-LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2250-2266. doi: 10.11948/20220455

Citation:

Man Xu, Ruyun Ma, Ting Wang. NON-SPURIOUS SOLUTIONS OF DISCRETE MIXED BOUNDARY VALUE PROBLEM WITH SINGULAR ϕ-LAPLACIAN[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 2250-2266. doi: 10.11948/20220455

NON-SPURIOUS SOLUTIONS OF DISCRETE MIXED BOUNDARY VALUE PROBLEM WITH SINGULAR ϕ-LAPLACIAN

1.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
2.
College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Author Bio: Email: ryma@xidian.edu.cn(R. Ma); Email: 17874158737@163.com(T. Wang)
Corresponding author: Email: xmannwnu@126.com(M. Xu)
Fund Project: The authors were supported by the NSFC (No. 11671322) and by the grant 20JR10RA100, 21JR1RA230, 2021A-006 and NWNU-LKQN2021-17

Abstract

Abstract

In this paper, we consider the differential and difference problems associated with the discrete approximation of classical radial solutions of the nonlinear Dirichlet problem for the prescribed mean curvature equation in Minkowski space
$ \begin{align} &-\mathrm{div}\Big(\frac{\text{grad} v}{\sqrt{1-|\text{grad} v|^2}}\Big) = f\Big(|x|,v,\frac{dv}{dr}\Big)\ \ \ \text{in}\ \mathcal{B},\\ &v = 0 \ \ \text{on}\ \partial \mathcal{B}, \end{align} $
where $ \mathcal{B} $ is the unit ball in $ \mathbb{R}^{N} $, div denotes the divergence operator of $ \mathbb{R}^{N} $, $ \text{grad}v $ is the gradient of $ v $, $ |\cdot| $ denotes the Euclidean norm in $ \mathbb{R}^{N} $, $ \frac{dv}{dr} $ stands for the radial derivative of $ v $ and $ f $ is a continuous function. By using lower and upper solutions, we prove the existence of solutions of the corresponding differential and difference problems, and based on the ideas of lower and upper $ \mu $-solutions show the solutions of the discrete problem can converge to the solutions of the continuous problem.
- Non-spurious solution /
- discrete boundary value problem /
- singular ϕ-Laplacian /
- lower and upper solutions /
- prescribed mean curvature equation
MSC: 34A45, 34B16, 35A01, 39A27

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References

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