{"ID":2829121,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.13664","arxiv_id":"2512.13664","title":"Exponential Absolute Minimizing extension and biased infinity Laplacian","abstract":"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.","short_abstract":"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|}...","url_abs":"https://arxiv.org/abs/2512.13664","url_pdf":"https://arxiv.org/pdf/2512.13664v1","authors":"[\"Yang Chu\"]","published":"2025-12-15T18:53:06Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.MG\",\"math.OC\",\"math.PR\"]","methods":"[]","has_code":false}