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<p>In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling-type equation Δ<sup>2</sup><i>u</i> + <i>λ</i>Δ<i>u</i> − <i>k</i><sup>2</sup><i>u</i> = 0 in a bounded analytic domain Ω ⊆ ℝ<sup><i>n</i></sup> with the homogeneous boundary conditions <i>u</i> = 0 and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({{\partial u} \over {\partial \nu}}=0\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>ν</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </math></EquationSource> </InlineEquation> on <i>∂</i>Ω, where <i>λ, k</i> are nonnegative real constants, and <i>ν</i> is the outer unit normal vector on <i>∂</i>Ω. We obtain that, the upper bounds for the maximal vanishing order of <i>u</i> and the <i>n</i> − 1-dimensional Hausdorff measure of the nodal set of <i>u</i> are both <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C({{\sqrt \lambda}} + {{\sqrt k}} + 1)\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow> <mrow> <msqrt> <mi>λ</mi> </msqrt> </mrow> </mrow> <mo>+</mo> <mrow> <mrow> <msqrt> <mi>k</mi> </msqrt> </mrow> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, where <i>C</i> is a positive constant only depending on <i>n</i> and Ω. Moreover, we also give a quantitative result of the propagation of smallness of <i>u</i>.</p>

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Some quantitative properties of solutions to the Buckling type equation

  • 期刊论文

  • 发表日期：

    2025年11月12日

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  • Long Tian,
  • Xiaoping Yang

摘要

In this paper, we investigate the quantitative unique continuation, propagation of smallness and measure bounds of nodal sets of solutions to the Buckling-type equation Δ2u + λΔu − k2u = 0 in a bounded analytic domain Ω ⊆ ℝn with the homogeneous boundary conditions u = 0 and \({{\partial u} \over {\partial \nu}}=0\) ∂ u ∂ ν = 0 on ∂Ω, where λ, k are nonnegative real constants, and ν is the outer unit normal vector on ∂Ω. We obtain that, the upper bounds for the maximal vanishing order of u and the n − 1-dimensional Hausdorff measure of the nodal set of u are both \(C({{\sqrt \lambda}} + {{\sqrt k}} + 1)\) C ( λ + k + 1 ) , where C is a positive constant only depending on n and Ω. Moreover, we also give a quantitative result of the propagation of smallness of u.