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 ${\hat{I}}^2$\textit{-Exponential Absolute Minimizing Extension} (${\hat{I}}^2$--AM) on arbitrary length space, which absolutely minimizing the exponential slope $$L_u^2(E):={\hat{I}}^2\underset{x,y\in E}{\sup }\frac{u(y)-e^{-{\hat{I}}^2|x-y|}u(x)}{1-e^{-{\hat{I}}^2|x-y|}}.$$We also define the corresponding Exponential McShane-Whitney-type extension, and ${\hat{I}}^2$-biased convexity, which equivalently characterize ${\hat{I}}^2$-AM and may be of independent interest. These generalize the classical Absolute Minimizing Lipschitz Extension as a special case when ${\hat{I}}^2=0$. 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 $\hat{I}{”}_{\infty }^{2}u=0$. We show that ${\hat{I}}^2$-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.

37 pages