Asymptotic behavior of solutions to elliptic equations in 2D exterior domains

TL;DR

This paper investigates the asymptotic behavior of solutions to second order elliptic equations in 2D exterior domains, providing near-sharp pointwise estimates at infinity under specific space conditions.

Contribution

It extends decay estimates for elliptic solutions in exterior domains by applying Lorentz and weak Lebesgue space conditions, building on methods from Navier-Stokes vorticity analysis.

Findings

Established pointwise decay estimates for solutions in Lorentz and weak Lebesgue spaces.

Connected decay properties of elliptic solutions to techniques used in Navier-Stokes vorticity studies.

Provided conditions under which solutions exhibit almost sharp asymptotic behavior.

Abstract

The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $L^{p,q}$ or the weak Lebesgue space $L^{p,\infty}$ with certain conditions on the coefficients, we give natural and an almost sharp pointwise estimate of the solution at spacial infinity. The proof is based on the argument by Korobkov--Pileckas--Russo [4], in which the decay property of the solution to the vorticity equation of the two-dimensional Navier--Stokes equations was studied.