2021/11/04 by Niklas L. P. Lundström, Lundström, Niklas L. P., Jesper Singh +1 · 1 citation
Computer Science · #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.2111.02721
Suppose that p ∈ (1,∞], ν∈ [1/2,∞), Sν= \ (x1,x2) ∈ ℝ2 ∖ \(0, 0)\: |ϕ| < \fracπ2ν\, where ϕ is the polar angle of (x1,x2). Let R>0 and ωp(x) be the p-harmonic measure of ∂ B(0,R) ∩ Sν at x with respect to B(0, R)∩ Sν. We prove that there exists a constant C such that C-1((|x|)/(R))k(ν,p) ≤ ωp(x) ≤ C ((|x|)/(R))k(ν,p) whenever x∈ B(0,R) ∩ S2ν and where the exponent k(ν,p) is given explicitly as a function of ν and p. Using this estimate we derive local growth estimates for p-sub- and p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmen-Lindelöf theorem for p-subharmonic functions in the unbounded sector Sν. Moreover, if p = ∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in ℝn. Finally, when ν∈ (1/2, ∞) and p∈ (1,∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on \partialSν.