2019/05/16 by Hynd, Ryan, Seuffert, Francis
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.07060
We study the limiting behavior as |x|→ ∞ of extremal functions u for Morrey's inequality on ℝn. In particular, we compute the limit of u(x) as |x|→ ∞ and show |x||Du(x)| tends to 0. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form -Δpu=c(δx0-δy0) for some c∈ ℝ and distinct x0,y0∈ ℝn. More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded p-harmonic functions on exterior domains of ℝn for p>n.