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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.19410 |
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| _version_ | 1866910764856508416 |
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| author | del Teso, Félix Rossi, Julio D. |
| author_facet | del Teso, Félix Rossi, Julio D. |
| contents | We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the $p$-Laplacian operator for $p>2$. Specifically, we characterize viscosity solutions to the $p$-Laplace equation $Δ_p u:=\nabla\cdot(|\nabla u|^{p-2} \nabla u) = f$ with a nontrivial right-hand side $f$, through novel asymptotic mean value formulas. While asymptotic mean value formulas for the homogeneous case ($f = 0$) have been previously established, leveraging the normalization $Δ_p^{\text{N}}u:=|\nabla u|^{2-p} Δ_p u = 0$, which yields the 1-homogeneous normalized $p$-Laplacian, such normalization is not applicable when $f \neq 0$. Furthermore, the mean value formulas introduced here motivate, for the first time in the literature, a game-theoretical approach for non-homogeneous $p$-Laplace equations. We also analyze the existence, uniqueness, and convergence of the game values, which are solutions to a dynamic programming principle derived from the mean value property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Game theoretical asymptotic mean value properties for non-homogeneous $p$-Laplace problems del Teso, Félix Rossi, Julio D. Analysis of PDEs Probability We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the $p$-Laplacian operator for $p>2$. Specifically, we characterize viscosity solutions to the $p$-Laplace equation $Δ_p u:=\nabla\cdot(|\nabla u|^{p-2} \nabla u) = f$ with a nontrivial right-hand side $f$, through novel asymptotic mean value formulas. While asymptotic mean value formulas for the homogeneous case ($f = 0$) have been previously established, leveraging the normalization $Δ_p^{\text{N}}u:=|\nabla u|^{2-p} Δ_p u = 0$, which yields the 1-homogeneous normalized $p$-Laplacian, such normalization is not applicable when $f \neq 0$. Furthermore, the mean value formulas introduced here motivate, for the first time in the literature, a game-theoretical approach for non-homogeneous $p$-Laplace equations. We also analyze the existence, uniqueness, and convergence of the game values, which are solutions to a dynamic programming principle derived from the mean value property. |
| title | Game theoretical asymptotic mean value properties for non-homogeneous $p$-Laplace problems |
| topic | Analysis of PDEs Probability |
| url | https://arxiv.org/abs/2412.19410 |