A Projected Tug-of-War Game for the Regularized $p$-Laplacian
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915974066733056 |
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| author | Ramezanzadeh, Behrooz Moosavi |
| author_facet | Ramezanzadeh, Behrooz Moosavi |
| contents | We give a tug-of-war interpretation of the regularized $p$-Laplacian $\divgg\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $Ω\subset\R^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $Δ_p w=0$ in $\R^{n+1}$. Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\R^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_01218 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Projected Tug-of-War Game for the Regularized $p$-Laplacian Ramezanzadeh, Behrooz Moosavi Analysis of PDEs We give a tug-of-war interpretation of the regularized $p$-Laplacian $\divgg\big((1+|Dv|^2)^{p/2-1}Dv\big)=0$ in a bounded domain $Ω\subset\R^n$, $p\ge 2$. The key is the linear lift $w(x,x_{n+1})=v(x)+x_{n+1}$, which identifies this equation with $Δ_p w=0$ in $\R^{n+1}$. Projecting the standard $(n+1)$-dimensional $p$-harmonious scheme onto $\R^n$ yields a discrete dynamic programming principle for which we prove existence, uniqueness, and Borel measurability of solutions with strip boundary data, identify the unique fixed point with the value of the projected game, and establish convergence to the viscosity solution as $\varepsilon\to 0$. |
| title | A Projected Tug-of-War Game for the Regularized $p$-Laplacian |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2605.01218 |