Solutions of Aronsson equation near isolated points

Abstract

When n ≥ 2, we show that for a non-negative solution of the Aronsson equation \({\fancyscript{A}_H (u) = D_x H(Du(x)) \, \cdot \, H_p(Du(x)) = 0}\) an isolated singularity x0 is either a removable singularity or \({u(x) = b + C_{k}^{H}(x-x_0) + o(|x - x_0|)}\) (or \({u(x) = b - C_{k}^{\hat{H}}(x-x_0) + o(|x - x_0|)}\)) for some k > 0 and \({b \in \mathbb{R}}\) . Here \({C_{k}^{H}\, {\rm and}\, C_{k}^{\hat{H}}}\) are general cone functions. This generalizes the asymptotic behavior theory for infinity harmonic functions by Savin et al. [13] (Int Math Res Not 2008: 23 pp, 2008). The Hamiltonian \({H \in C^2(\mathbb{R}^n)}\) is assumed to be non-negative and uniformly convex.