Exponential Absolute Minimizing extension and biased infinity Laplacian
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arXiv
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| Format: | Preprint |
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2025
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| author | Chu, Yang |
| author_facet | Chu, Yang |
| contents | We study the variational structure of the biased infinity Laplacian by introducing a notion of the $β$\textit{-Exponential Absolute Minimizing Extension} ($β$--AM) on arbitrary length space, which absolutely minimizing the exponential slope
$$ L^β_u (E) := β\sup_{x,y \in E} \frac{u(y) - e^{-β|x-y|} u(x)}{1- e^{-β|x-y|}}. $$We also define the corresponding Exponential McShane-Whitney-type extension, and $β$-biased convexity, which equivalently characterize $β$-AM and may be of independent interest. These generalize the classical Absolute Minimizing Lipschitz Extension as a special case when $β= 0$. In Euclidean space with Euclidean norm, this corresponds to the Aronsson equation with Hamiltonian \[ H(u, \nabla u) = |\nabla u| + βu, \] equivalently viscosity solutions of $Δ_{\infty}^β u = 0$. We show that $β$-AM arises as the continuum value of a biased tug-of-war game. Analogous to the unbiased case, we derive various properties of this extension. As an application, we further show that the linear blow-up property holds for biased infinity harmonic functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_13664 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exponential Absolute Minimizing extension and biased infinity Laplacian Chu, Yang Analysis of PDEs Metric Geometry Optimization and Control Probability We study the variational structure of the biased infinity Laplacian by introducing a notion of the $β$\textit{-Exponential Absolute Minimizing Extension} ($β$--AM) on arbitrary length space, which absolutely minimizing the exponential slope $$ L^β_u (E) := β\sup_{x,y \in E} \frac{u(y) - e^{-β|x-y|} u(x)}{1- e^{-β|x-y|}}. $$We also define the corresponding Exponential McShane-Whitney-type extension, and $β$-biased convexity, which equivalently characterize $β$-AM and may be of independent interest. These generalize the classical Absolute Minimizing Lipschitz Extension as a special case when $β= 0$. In Euclidean space with Euclidean norm, this corresponds to the Aronsson equation with Hamiltonian \[ H(u, \nabla u) = |\nabla u| + βu, \] equivalently viscosity solutions of $Δ_{\infty}^β u = 0$. We show that $β$-AM arises as the continuum value of a biased tug-of-war game. Analogous to the unbiased case, we derive various properties of this extension. As an application, we further show that the linear blow-up property holds for biased infinity harmonic functions. |
| title | Exponential Absolute Minimizing extension and biased infinity Laplacian |
| topic | Analysis of PDEs Metric Geometry Optimization and Control Probability |
| url | https://arxiv.org/abs/2512.13664 |