A Note on Non-Negative Singular Infinity-Harmonic Functions in the Half-Space

Abstract

In this work we study non-negative singular infinity-harmonic functions in the half-space. We assume that solutions blow-up at the origin while vanishing at infinity and on a hyperplane. We show that blow-up rate is of the order |x| −1/3 .

A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results

“…For r > 0, let M u (r) = sup z∈∂Br (o)∩Cα u(z). We extend the result in [7] to show optimality of the Aronsson singular examples [6]. Theorem 1.3.…”

“…The last conclusion in Theorem 1.3 will follow from the works [6,7]. While Theorem 1.3 applies to special situations, the main purpose is to understand better the blowup rates of singular solutions, and in some situations decay rates.…”

“…A decay rate and a halving property for such functions on the half-space will also be presented. Another application will be to show optimality of Aronsson's singular examples in cones, thus generalizing the result in [6,7].…”

Separable infinity harmonic functions in cones

“…Its existence is due to Aronsson [2]. The corresponding circular function, ω(σ) = (cos σ) 4 3 − (sin σ) 4 3 , admits four nodal sets on S 1 . When k = 1, then β 1 = 1 3 .…”

“…When k = 1, then β 1 = 1 3 . It is proved in [4] that any positive infinity harmonic function in a half-space which vanishes on the boundary except at one point blows-up like the separable infinity harmonic function u(r, σ) = r − 1 3 ψ(σ).…”