Solutions of Aronsson equation near isolated points

Julin, Vesa
March 2010
Calculus of Variations & Partial Differential Equations;Mar2010, Vol. 37 Issue 3/4, p303
Academic Journal
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.


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