Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929606857064448 |
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| author | Han, Jeongmin |
| author_facet | Han, Jeongmin |
| contents | In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} Δ_{p}^{N}u=0 & \textrm{in $ Ω$,}\\ \langle β, Du \rangle + γu = γG & \textrm{on $ \partial Ω$,}\\ \end{array} \right. \end{align*} where $Δ_{p}^{N}$ is the normalized $p$-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using `shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian Han, Jeongmin Analysis of PDEs 91A05, 91A15, 35D40, 35B65 In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} Δ_{p}^{N}u=0 & \textrm{in $ Ω$,}\\ \langle β, Du \rangle + γu = γG & \textrm{on $ \partial Ω$,}\\ \end{array} \right. \end{align*} where $Δ_{p}^{N}$ is the normalized $p$-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using `shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence. |
| title | Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian |
| topic | Analysis of PDEs 91A05, 91A15, 35D40, 35B65 |
| url | https://arxiv.org/abs/2405.14568 |