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<p>The asymptotic behavior of solutions to second order elliptic equations in two-dimensional exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^{p,q}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> or the weak Lebesgue space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^{p,\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>∞</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> with certain conditions on the coefficients, we give a natural and the optimal sharp pointwise estimate of the solution at spatial infinity. The proof is based on the level set approach of solutions introduced by Korobkov et al. (Arch Rat Mech Ann 233:385–407, 2019), in which the decay property of the solution to the vorticity equation of the two-dimensional Navier–Stokes equations was studied.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月12日

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  • Hideo Kozono,
  • Yutaka Terasawa,
  • Yuta Wakasugi

摘要

The asymptotic behavior of solutions to second order elliptic equations in two-dimensional exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space \(L^{p,q}\) L p , q or the weak Lebesgue space \(L^{p,\infty }\) L p , ∞ with certain conditions on the coefficients, we give a natural and the optimal sharp pointwise estimate of the solution at spatial infinity. The proof is based on the level set approach of solutions introduced by Korobkov et al. (Arch Rat Mech Ann 233:385–407, 2019), in which the decay property of the solution to the vorticity equation of the two-dimensional Navier–Stokes equations was studied.