指数绝对最小化扩展与有偏无穷拉普拉斯算子
Exponential Absolute Minimizing extension and biased infinity Laplacian
AI总结：
本文研究了有偏无穷拉普拉斯算子的变分结构，引入β-指数绝对最小化扩展概念，并证明其与有偏拉普拉斯博弈的联系。
摘要：
我们通过在任意长度空间上引入β-指数绝对最小化扩展（β--AM）的概念，研究了有偏无穷拉普拉斯算子的变分结构。该概念绝对最小化指数斜率$$ L^β_u (E) := β\sup_{x,y \in E} \frac{u(y) - e^{-β|x-y|} u(x)}{1- e^{-β|x-y|}}.$$我们还定义了相应的指数McShane-Whitney型扩展以及β-有偏凸性，这些等价地刻画了β-AM并可能具有独立价值。这些概念作为β=0时的特殊情况，扩展了经典的绝对最小化Lipschitz扩展。在欧几里得空间中，这对应于哈密顿量为$$ H(u, \nabla u) = |\nabla u| + βu $$的Aronson方程，等价于无穷拉普拉斯方程的粘性解。我们证明β-AM出现在有偏拉普拉斯博弈的连续值中。类似于无偏情况，我们推导了该扩展的各种性质。作为应用，我们进一步证明有偏无穷调和函数具有线性发散性质。
英文摘要：
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.
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